PRINCIPLES 

OF 

REINFORCED  CONCRETE 
CONSTRUCTION 


BY 

F.    E.   TURNEAURE 

Dean  of  the  College  of  Engineering    University  of  Wisconsin 
AND 

-*'         E.   R.   MAURER 

Professor  of  Mechanics,  University  of  Wisconsin 


SECOND    EDITION,  REVISED    AND    ENLARGED 
TOTAL  ISSUE,    SIX    THOUSAND 


NEW    YORK 

JOHN  WILEY  &  SONS 

London:    CHAPMAN  &   HALL,    Limited 

1909. 


GENERAL 


Copyright,  1907,  1909 

JST 
F.  E.  TURNEAURE  AND  E.  R.  MAURER 


ISubcrl  Drummonli  anil  (Uom|iang 


PREFACE  TO  THE  SECOND  EDITION. 


IN  the  preparation  of  this  edition  numerous  changes  have 
been  made  and  a  large  amount  of  material  has  been  added. 
Results  of  recent  important  experiments  have  been  included 
and  the  analytical  treatment  has  been  considerably  extended 
in  certain  directions.  New  experimental  data  are  noted  in 
nearly  every  division  o:  the  subject,  but  of  especial  import- 
ance are  the  results  on  bond  strength,  the  strength  of  beams 
in  shear,  and  the  strength  of  columns.  The  general  subject 
of  the  deflection  of  beams  has  been  treated  at  considerable 
length,  both  theoretically  and  experimentally.  The  treat- 
ment of  T-beams  has  been  extended  and  several  diagrams 
have  been  added  in  Chapter  V  for  use  in  designing  this  form 
of  beam.  The  chapter  pertaining  to  working  stresses  and 
general  constructive  details  has  been  largely  rewritten,  especial 
attention  being  given  to  the  subject  of  shear  reinforcement. 
In  Chapter  VII  the  treatment  of  continuous  beams  has  been 
considerably  amplified.  In  the  chapter  on  arches  the  methods 
of  calculation  have  been  more  fully  explained  by  means  of 
an  additional  example,  fully  worked  out,  in  which  use  is  made 
of  influence  lines  for  fiber  stress.  A  chapter  has  been  added 
on  chimneys,  which  includes  a  fairly  complete  analytical  treat- 
ment of  this  subject.  It  is  believed  that  the  changes  and 
additions  which  have  been  made  will  measurably  enhance  the 
usefulness  of  the  work. 

F.  E.  T. 

E.  R.  M. 

MADISON,  Wis., 
April,  1909. 


m 


185030 


PREFACE  TO  THE  FIRST  EDITION. 


IN  the  present  volume  the  authors  have  endeavored  to 
cover,  in  a  systematic  manner,  those  principles  of  mechanics 
underlying  the  design  of  reinforced  concrete,  to  present  the 
results  of  all  available  tests  that  may  aid  in  establishing  coeffi- 
cients and  working  stresses,  and  to  give  such  illustrative 
material  from  actual  designs  as  may  be  needed  to  make  clear 
the  principles  involved. 

The  work  is  essentially  divided  into  two  parts:   Chapters 

I  to  VI  treat  of  the  theory  of  the  subject  and  the  results  of 
experiments,  while  the  remaining  chapters  treat  of  the  use  of 
reinforced  concrete  in  various  forms  of  structures.    In  Chapter 

II  the  properties  of  plain  concrete  and  of  steel  are  considered 
to  a  sufficient  extent  to  give  accurate  notions  of  their  relation 
to  the  general  subject  in  hand.    The  subjects  of  adhesion  and 
of  relative  contraction  and  expansion  are  also  discussed  in  this 
chapter.    In  Chapter  III  is  given  a  full  theoretical  treatment 
of  reinforced  concrete,  avoiding  so  far  as  possible  empirical 
rules  and  methods;  and  in  Chapter  IV  are  presented  the  most 
important  available  tests  on  beams  and  columns,  analyzed  and 
correlated,  so  far  as  may  be,  with   reference  to   theoretical 
principles.    The  subjects  of  working  stresses  and  economical 
proportions  are  considered  in  Chapter  V.    In  Chapter  VI  are 
brought   together  in   convenient    form   all  the  formulas  and 
diagrams  needed  for  practical  use.     There  are  also  included 
tables  relating  to  reinforcing  bars  and  a  comprehensive  table 


vi  PREFACE 

of  the  strength  of  floor  slabs.  This  chapter  is,  for  most  pur- 
poses, complete  in  itself,  so  that  the  reader  need  not  refer  to 
any  other  portion  of  the  work  in  order  to  use  it  in  designing. 

Following  the  theoretical  portions  are  chapters  on  the 
application  of  reinforced  concrete  to  building  construction, 
arches,  retaining  walls,  dams,  and  miscellaneous  structures. 
In  these  chapters  the  analysis  of  various  features  is  given, 
where  the  use  of  reinforced  concrete  involves  problems  new 
and  unfamiliar.  A  complete  general  analysis  of  the  solid  arch 
rib  is  also  given,  which,  the  authors  believe,  offers  advantages 
over  the  usual  graphical  method.  It  is  primarily  an  analytical 
method,  but  may  be  shortened  by  obvious  simple,  graphical 
aids.  Stresses  in  the  concrete  and  steel  are  readily  calculated 
by  the  use  of  diagrams  in  Chapter  VI.  In  the  chapters  on  the 
application  of  reinforced  concrete  it  has  not  been  the  aim  to 
cover  practical  construction  in  all  its  phases;  for  this  the 
reader  is  referred  to  the  more  voluminous  works  on  the  subject. 
It  is  hoped,  however,  that  as  a  treatment  of  the  principles  of 
design  the  work  may  prove  of  service  to  the  student  and  the 
engineer. 

F.   E.   TURNEAURE. 

E.  R.  MAURER. 

MADISON,  Wis.,  Sept.,  1007. 


CONTENTS. 


CHAPTER  I. 

PAQH 

INTRODUCTORY       .        .......  ~      »        .        .        .  1 

Historical  Sketch.     Use  and  Advantages  of  Reinforced  Concrete. 


.  .  CHAPTER    II. 
PROPERTIES  OP  THE  MATERIAL    ...        .        .        .        8 

Concrete:  General  Requirements.  Cement.  Sand.  Broken 
Stone  or  Gravel.  Proportion  of  Ingredients.  Consistency. 
Compressive  Strength.  Tensile  Strength.  Transverse  Tests, 
Shearing  Strength.  Elastic  Properties.  Stress-strain  Curve  in 
Compression.  Modulus  of  Elasticity.  Elastic  Limit.  Stress- 
strain  Curves.  Coefficient  of  Expansion.  Contraction  and 
Expansion.  Weight.  Properties  of  Cinder  Concrete.  Rein- 
forcing  Steel:  General  Requirements.  Special  Forms  of  Bars. 
Quality  of  the  Material.  Tensile  Strength.  Modulus  of  Elas- 
ticity. Elastic  Elongation.  Coefficient  of  Expansion.  Prop- 
erties of  Concrete  and  Steel  in  Combination:  Bond  Strength. 
Mechanical  Bond.  Ratio  of  Moduli  of  Elasticity.  Tensile  Stress 
and  Elongation  of  Concrete  when  Reinforced.  Relative  Con- 
traction and  Expansion. 

CHAPTER   III. 
GENERAL  THEORY       ..        .        .        .        .        .        .-      -.      47 

Kinds  of  Members.     Relation  of  Stress  Intensities  in  Concrete 
and  Steel.     Distribution  of  Stress  in  a  Homogeneous  Beam. 

vii 


viii  CONTENTS. 

PAQB 

Purpose  and  Arrangement  of  Steel  Reinforcement.  The  Com- 
mon Theory  of  Flexure  and  its  Modification  for  Concrate. 
Resisting  Moment  and  Inefficiency  of  Concrete  Beams.  Varieties 
of  Flexure  Formulas.  Flexure  Formulas  for  Working  Loads. 
Flexure  Formulas  for  Ultimate  Loads.  Flexure  Formulas  for 
any  Load  up  to  Ultimate.  Comparison  of  Flexure  Formulas. 
Flexure  Formulas  for  T-Beams.  Beams  Reinforced  for  Com- 
pression. Flexure  and  Direct  Stress.  Shearing  Stresses  in  Re- 
inforced Beams.  Bond  Stress.  Deflection  of  Beams.  Strength 
of  Columns. 


CHAPTER   IV. 
TESTS  OF  BEAMS  AND  COLUMNS 135 

Beams :  Methods  of  Failure  of  a  Reinforced  Concrete  Beam. 
Tests  of  Beams  giving  Steel-tension  Failures.  Position  of 
Neutral  Axis  and  Value  of  n.  Observed  and  Calculated 
Stresses  in  the  Steel.  Compressive  Stresses  in  Concrete.  Con- 
clusions Regarding  Moment  Calculations.  Tests  in  which 
Failure  Occurred  by  Diagonal  Tension.  Methods  of  Web  Rein- 
forcement. Action  of  Web  Reinforcement.  Effect  of  Stirrups. 
Results  of  Tests.  Tests  on  T-Beams.  Conclusions  as  to 
Shearing  Strength.  Beams  Reinforced  in  Compression.  Tests 
on  Deflections.  Columns:  Plain  Concrete  Columns.  Tests  on 
Columns  with  Longitudinal  Reinforcement.  Hooped  Concrete 
Columns.  Fatigue  Tests. 

CHAPTER   V. 
WORKING  STRESSES  AND  GENERAL  CONSTRUCTIVE  DETAILS    208 

Working  Stresses  and  Factors  of  Safety.  Relative  Effect  of 
Dead  and  Live  Loads.  Beams :  Working  Formulas.  Working 
Stresses  in  Concrete  and  Steel.  Quality  of  Steel.  Bond  Stress. 
Shearing  Stresses.  Calculation  of  Web  Reinforcement.  Spacing 
of  Bars  Economical  Proportions  and  Working  Stresses.  T- 
Beams.  Columns:  Working  Stresses.  Long  Columns.  Column 
Details.  Economy  in  the  Use  of  Reinforced  Columns.  Dura- 
bility of  Reinforced  Concrete:  Protection  of  Steel  from  Corrosion. 
Fireproofing  Effect  of  Concrete.  Reinforcing  Against  Shrinkage 
and  Temperature  Cracks. 


CONTENTS.  ix 

CHAPTER   VI. 

PAGE 

FORMULAS,  DIAGRAMS  AND  TABLES     .        ;        .       v-     .     259 
Rectangular  Beams.     T-Beams.     Beams  Reinforced  for  Com- 
pression.    Flexure    and    Direct    Stress.     Shearing    and    Bond 
Stress.     Columns.  -~  Shrinkage  and  Temperature  Stresses.     Co- 
efficients and  Working  Stresses.     Diagrams.      Tables. 


CHAPTER   VIL* 

BUILDING  CONSTRUCTION      .        .        .      . ..  ."':.:.     .;  . ....    301 

Division  of  the  Subject.  General  Arrangement  of  Concrete 
Floors.  Stresses  in  Continuous  Beams.  Effect  of  Rigid  Sup- 
ports. Slabs  Reinforced  in  Two  Directions.  Reinforcement  to 
Prevent  Cracks.  Floor  Slabs  Supported  on  Steel  Beams. 
Floor  Slabs  in  All-Concrete  Construction.  Beams  and  Girders. 
Columns.  Examples  of  Floor  and  Column  Design.  Footings. 
Walls  and  Partitions. 


CHAPTER  VIII. 
ARCHES        .        . .    333 

Advantages  of  the  Reinforced  Arch.  Methods  of  Reinforce- 
ment. Analysis  of  the  Arch:  General  Method  of  Procedure. 
Thrust,  Shear,  and  Moment  at  the  Crown.  Thrust,  Shear,  and 
Moment  at  Any  Point.  Partial  Graphical  Calculation.  General 
Observations.  Division  of  Arch  Ring  to  give  Constant  ds/I. 
Temperature  Stresses.  Stresses  Due  to  Shortening  of  Arch  from 
Thrust.  Deflection  of  the  Crown.  Unsymmetrical  Arches. 
Applications.  Maximum  Stresses  in  the  Arch  Ring.  Illustra- 
tive Examples  of  Design. 


CHAPTER    IX. 

RETAINING- WALLS  AND  DAMS      .        .        .        .        .        .     370 

Advantages  of  Reinforced  Concrete.  Retaining  Walls :  Method 
of  Determining  Stability.  Equivalent  Fluid  Pressure  for 
Ordinary  Masonry  Walls.  Stability  of  Reinforced  Concrete 
Walls.  Design  of  Wall.  Illustrative  Examples.  Rect- 
angular Walls  Supported  at  the  Top.  Dams:  Stability  and 
Examples. 


x  CONTENTS. 

CHAPTER  X. 

PA«E 

MISCELLANEOUS  STRUCTURES    .      y       .       »       .  :  .-  385 

Simple  Beam  Bridges.  Concrete  Trestles.  Pipe  and  Box 
Culverts:  The  Circular  Culvert.  The  Rectangular  Culvert. 
Arrangement  of  Reinforcement.  Tests  on  Pipe.  Illustrative 
Examples.  Conduits  and  Pipe  Lines.  Tanks,  Reservoirs, 
Bins,  etc. 

CHAPTER  XI. 

REINFORCED-CONCRETE  CHIMNEYS 398 

General  Description.  Design.  Wind  Stresses.  Brick  Chimneys. 
Temperature  Stresses.  Chimney  Temperatures.  Bases. 


REINFORCED-CONCRETE   CONSTRUCTION. 


CHAPTER  I. 

INTRODUCTORY. 

i.  Historical  Sketch. — The  invention  of  reinforced  con- 
crete is  usually  credited  to  Joseph  Monier.  but  his  first  con- 
structions are  antedated  by  those  of  Lambgt^  who  in  1850 
constructed  a  small  boat  of  reinforced  concrete  and  in  1855 
exhibited  the  same  at  the  Paris  Exposition.  In  this  latter  year 
Lambot  took  out  patents  on  this  form  of  construction;  it  was 
regarded  by  him  as  especially  well  adapted  to  shipbuilding, 
reservoir  work,  etc. 

In  1861,  Monier,  who  was  a  Parisian  gardener,  constructed 
tubs  and  tanks  of  concrete  surrounding  a  framework  or  skeleton 
of  wire.  In  the  same  year  Coignet  announced  his  principles 
for  reinforcing  concrete,  and  proposed  construction  of  beams, 
arches,  pipes,  etc.  Both  he  and  Monier  executed  some  work  in 
the  new  material  at  the  Paris  Exposition  of  1867.  In  this  year 
Monier  took  out  patents  on  his  reinforcement.  It  consists  of 
two  sets  of  parallel  bars,  one  set  at  right  angles  to  and  lying 
upon  the  other,  thus  forming  a  mesh  of  bars.  This  system,  and 
slight  modifications  of  it,  are  extensively  used  at  the  present 
time,  particularly  for  slab  reinforcement.  Though  even  the 
early  Monier  patents  covered  principles  of  wide  application, 
still  the  early  work  in  reinforced  concrete  was  confined  to  a 
comparatively  narrow  field. 


2  INTRODUCTORY.  [On.  I. 

In  1884-5  the  German  and  American  rights  of  the  Monier 
patents  fell  into  the  hands  of  German  engineers.  One  of  these, 
G.  A.  Wayss,  and  J.  Bauschinger  at  once  began  an  experimental 
investigation  of  the  Monier  system,  and  in  1887  they  published 
their  findings.  The  investigation  proved  reinforced  concrete  a 
valuable  means  of  construction,  and  furnished  some  formulas 
and  methods  for  design.  From  this  time  on,  the  use  of  re- 
inforced concrete  in  Austria  spread  rapidly,  and  a  few  years 
ago  the  engineers  of  that  country  were  credited  with  having 
done  more  to  develop  the  new  construction  than  those  of  any 
other  country.  Among  these  engineers  should  be  mentioned 
Melan,  who  in  the  early  90's  originated  a  system  in  which  I  or 
T  beams  are  the  principal  element  of  strength,  providing  com- 
pressive  as  well  as  tensile  strength.  In  Germany  government 
regulations  hindered  the  application  of  reinforced  concrete  for  a 
time,  but  now  it  is  widely  used  in  that  country.  Over  two 
hundred  systems  of  reinforcement,  it  has  been  stated,  have 
been  developed  in  Germany  alone. 

In  France  the  Monier  system  was  never  developed  as  in 
countries  already  mentioned.  Here,  as  elsewhere,  many  other 
systems  of  reinforcement  were  invented  from  time  to  time, 
among  which  should  be  mentioned  that  of  Hennebique,  who 
was  probably  the  first  to  use  stirrups  and  "bent-up"  bars. 
This  system  is  in  general  use,  and  the  elements  of  Hennebique's 
system  are  probably  more  widely  used  than  those  of  any 
other. 

In  England  and  America  the  first  use  of  iron  or  steel  with 
concrete  arose  in  the  effort  to  fireproof  the  former  by  means  of 
the  latter.  Attempting  to  utilize  also  the  strength  of  concrete, 
Hyatt  built  beams  of  concrete  reinforced  with  metal  in  various 
ways,  and  with  Kirkaldy  of  London  performed  tests  on  such 
beams  and  published  the  results  of  the  investigation  in  1877. 
The  first  reinforced-concrete  work  in  the  United  States  was  done 
in  1875  by  W.  E.  Ward,  who  constructed  a  building  in  New  York 
state  in  which  walls,  floor-beams,  and  roof  were  made  of  con- 
crete reinforced  with  metal  to  provide  tensile  strength.  But 


I  1.]  HISTORICAL  SKETCH.  3 

the  Pacific  Coast  saw  the  actual  early  development  of  this  form 
of  construction.  H.  P.  Jackson,  G.  W.  Percy,  and  E.  L.  Ran- 
some  were  the  pioneer  workers.  Jackson  has  been  credited  with 
reinforced  constructions  dating  as  far  back  as  1877,  but  Ransome 
executed  the  most  notable  early  examples.  Among  these  are  a 
warehouse  (1884  or  '85),  a  factory  building  a  few  years  later,  the 
building  of  the  California  Academy  of  Science  (1888  or  '89), 
and  the  museum  building  of  Leland  Stanford  Junior  University 
(1892).  Percy  was  the  architect  of  the  last  two.  The  museum 
building  contains  spans  of  45  feet  and  is  reinforced  through- 
out. This  and  the  Academy  building  withstood  the  recent 
earthquake  remarkably  well — the  museum  better  than  its  two 
brick  annexes. 

Other  pioneer  constructors  in  reinforced  concrete  in  this 
country  were  F.  von  Emperger  and  Edwin  Thacher.  The 
former  introduced  the  Melan  system  (1894)  and  built  the  first 
reinforced  arch  bridges  of  considerable  span.  Thacher  also 
was — and  still  is — a  bridge-builder.  His  first  large  reinforced- 
concrete  bridge  was  built  in  1896  and  was  without  precedent 
here  or  in  Europe. 

America  is  the  home  of  the  " patent  bar".  Both  Ransome 
and  Thacher  invented  bars  known  by  their  respective  names, 
the  patented  feature  of  which  is  to  furnish  a  "grip"  between 
bar  and  concrete;  besides  these  two  there  are  several  others  on 
the  market  designed  to  give  additional  grip  or  bond.  There  are 
also  patented  bars  for  supplying  " shear  reinforcement".  Some 
of  these  forms  have  been  introduced  into  Europe. 

Reinforced-concrete  construction  has  had  a  remarkable 
development,  particularly  in  the  last  decade,  and  is  now  re- 
garded by  engineers  and  architects  generally  as  a  safe  form  of 
construction  with  a  wide  field  of  economical  application.  Com- 
mon practice  has  already  established  itself  in  some  directions, 
and  rational  principles  are  available  for  much  design  work. 
Outstanding  uncertainties  are  under  investigation  in  many 
quarters,  and  the  time  is  not  far  distant  when  "good  practice" 
in  reinforced  concrete  will  have  been  established. 


4  INTRODUCTORY.  [Cn.  L 

2.  Use  and  Advantages  of  Reinforced  Concrete.— A  com- 
bination of  steel  and  concrete  constitutes  a  form  of  construc- 
tion possessing  to  a  large  degree  the  advantages  of  both  mate- 
rials without  their  disadvantages.  It  will  be  desirable  at  the 
outset  to  consider  briefly  these  advantages  in  order  better  to 
appreciate  the  field  in  which  this  type  of  construction  is  likely 
to  be  most  successful. 

Steel  is  a  material  especially  well  suited  to  resist  tensile 
stresses,  and  for  such  purposes  the  most  economical  form — 
the  solid  compact  bar — is  well  adapted.  To  resist  compressive 
stresses  steel  must  be  made  into  more  expensive  forms,  con- 
sisting of  relatively  thin  parts  widely  spread,  in  order  to  provide 
the  necessary  lateral  rigidity.  A  serious  disadvantage  in  the 
use  of  steel  in  many  locations  is  its  lack  of  durability;  and, 
again,  a  comparatively  low  degree  of  heat  destroys  its  strength, 
thus  rendering  it  necessary  to  add  a  protective  covering  where 
a  fire-proof  structure  is  demanded.  Steel  is  a  relatively  expen- 
sive building  material,  and  its  cost  tends  to  increase. 

Concrete  is  characterized  by  low  tensile  strength,  relatively 
high  compressive  strength,  and  great  durability.  It  is  a  good 
fire-proof  material,  and  therefore  serves  as  a  good  fire-proof 
covering  for  steel.  It  is  also  found  that  steel  well  covered  by 
concrete  is  thoroughly  protected  from  corrosion.  Concrete  is 
also  a  comparatively  cheap  material  and  is  readily  available 
in  almost  any  location. 

In  the  design  of  structural  members  these  qualities  of  steel 
and  concrete  will  lead  to  the  use  of  the  two  materials  about 
as  follows:  For  those  structural  members  carrying  purely  ten- 
sile stresses  steel  must  be  employed,  but  it  may  be  surrounded 
by  concrete  as  a  protection  against  corrosion  and  fire,  or  merely 
for  the  sake  of  appearance.  For  those  members  sustaining 
purely  compressive  stresses  concrete  is  fundamentally  the 
better  and  cheaper  material.  With  concrete  costing  30  cents 
per  cubic  foot,  for  example,  and  steel  4  cents  per  pound,  or 
about  $20.00  per  cubic  foot,  and  with  working  stresses  of 
400  and  15,000  lbs/in2,  respectively,  the  relative  cost  of  the 


§  2.]  USE  AND   ADVANTAGES.  5 

30    .  2000 

two  materials  for  carrying  a  given  load  is  as  -r^.  is  to  , 

4UU  -LOjUUU 

or  as  45  is  to  80.  For  large  and  compact  compressive  members 
plain  concrete  will  therefore  naturally  be  used,  especially  where 
durability  is  a  factor.  For  more  slender  members,  however, 
such  as  long  columns,  plain  concrete  is  too  brittle  a  material, 
and  therefore  too  much  affected  by  secondary  and  unknown 
stresses  to  be  satisfactory;  and  for  such  members  steel  alone, 
or  the  two  materials  in  combination,  will  preferably  be  used. 
Steel  may  be  used  with  concrete  in  the  form  of  small  rods 
to  reinforce  the  concrete;  or  it  may  be  used  in  larger  sec- 
tions and  simply  surrounded  and  held  rigidly  in  place  by 
the  concrete,  most  of  the  load  being  carried  by  the  steel;  or, 
finally,  a  steel  column  may  be  used  and  merely  fireproofed 
by  the  concrete.  As  the  cost  of  steel  in  the  form  of  rods  is 
much  less  than  in  the  form  of  built  members,  and  as  com- 
pressive stresses  can,  in  general,  be  carried  more  cheaply  by 
concrete  than  by  steel,  economical  construction  will  lead  to 
the  use  of  the  maximum  amount  of  concrete  and  the  minimum 
amount  of  steel  consistent  with  safety,  although  this  prin- 
ciple will  be  modified  by  various  practical  considerations. 

For  those  structural  forms  in  which  both  tension  and  com- 
pression exist,  that  is  to  say,  in  all  forms  of  beams,  the  com- 
bination of  the  two  materials  is  particularly  advantageous. 
Here  the  tensile  stresses  are  carried  by  steel  rods  embedded 
in  the  concrete  near  the  tension  side  of  the  beam.  The  steel 
is  thus  used  in  its  cheapest  form,  it  is  thoroughly  protected 
by  the  concrete,  and  the  compressive  stresses  are  carried  by 
the  concrete.  Concrete  alone  cannot  be  used  to  any  appre- 
ciable extent  to  carry  bending  stresses  on  account  of  its  low 
and  uncertain  tenacity,  but  a  concrete  beam  with  steel  rods 
embedded  in  it  to  carry  the  tensile  stresses  is  a  strong,  economical, 
and  very  durable  form  of  structure. 

From  these  considerations  it  follows  that  reinforced-con- 
crete  construction  is  advantageous  to  varying  degrees  in  dif- 
ferent types  of  structures.  Some  of  the  most  important  of 


6  INTRODUCTORY.  [Cn  L 

these  types  will  here  be  noted,  together  with  the  advan- 
tages accompanying  the  use  of  reinforced  concrete  in  their 
design. 

3.  Buildings. — This  type  of  construction  is  especially  useful 
for  floor-slabs  and  ttf  a  somewhat  less  degree  for  beams,  girders, 
and  columns.     It  is  also  well  adapted  for  footings  in  founda- 
tions, being  more  economical  than  I-beam  footings  embedded 
in  concrete. 

4.  Culverts  and  small  Girder  Bridges. — Very  satisfactory  on 
account  of  its  simplicity  and  economy  as  compared  to  masonry 
arches,   and  because  of    its  durability   as   compared  to  steel 
bridges. 

5.  Retaining-walls,  Dams,  and  Abutments. — Often  economical 
for  such  structures  as  compared  to  ordinary  masonry.     Plain 
masonry  structures  of  this  kind  are  designed  to  resist  lateral 
forces  by  their  weight  alone,  the  resulting  compressive  stresses, 
except  in  extremely  large  structures,   being  very  small  and 
much  below  safe  values.     By  the  use  of  reinforced  concrete 
these  structures  can  be  designed  of  a  more  economical  type 
and  so  arranged  as  to  utilize  the  concrete  in  the  form  of  beams, 
thus  developing  more  nearly  the  full  compressive  strength  of 
the  material.     The  steel  reinforcement  is  fully  protected  from 
corrosion,  a  factor  which  prevents  the  use  of  all-steel  frames 
for  structures  of  this  class. 

6.  Arch  Bridges. — In  this  form  of  structure  reinforced  con- 
crete possesses  less  advantage  over  ordinary  masonry  than  in 
those  forms  where  the  compressive  stresses  are  less  important. 
In  an  arch  the  stresses  are  principally  compressive,  and  these 
do  not  require  steel  reinforcement;    it  is  only  to  provide  for 
the  relatively  small  bending  stresses  due  to  moving  loads,  or 
as  a  precaution  against  undesirable  cracks,  that  steel  is  ser- 
viceable.   No  large  economy  can  be  obtained  through  its  use. 
By  reason  of  greater  simplicity  and  the  less  expensive  abutments 
required,  a  flat-top  culvert  or  beam  bridge,  with  abutments 
of  reinforced  concrete,  is  more  advantageous  for  short  spans 
than  the  arch. 


$  11.]  USE  AND  ADVANTAGES.  7 

7.  Reservoir  Walls,  Floors,  and  Roofs. — Very  well  adapted 
as  a  durable  material  and  lending  itself  to  lighter  design  than 
common  masonry. 

8.  Conduits  and  Pipe  Lines. — Reinforced  concrete  can  often 
be  used  to  great  advantage  in  a  water-conduit  or  large  sewer. 
It  is  also  sometimes  used  for  pipe  lines  and  tanks  under  pres- 
sure, the  steel  being  relied  upon  to  resist  the  tensile  stresses, 
while  the  concrete  serves  as  a  protection  and  as  a  water-tight 
covering.    The  amount  of  steel   may  thus  be  determined  by 
considerations  of  strength  alone,  where  otherwise  a  much  larger 
amount  of  metal  would  be  needed  and  in  a  more  expensive 
form. 

9.  Elevated  Tanks,  Bins,  etc. — Advantageous  because  of  its 
durability  and  its  adaptability  in  the  construction  of  heavy 
floors   and  walls   subjected   to   lateral   pressure.     Of   especial 
value  for  coal-bins,  either  for  flooring  and  lining  alone,  or  for 
the  entire  structure. 

10.  Chimneys  and  Towers. — Possesses  advantages  over  brick 
or  stone  masonry  in  the  fact  that  it  forms  a  structure  of  mono- 
lithic character,  resulting  in  greater   certainty  in  the  stresses 
and  economy  in  design. 

11.  Piles,  Railroad  Ties,  etc. — The  use  of  a  moderate  amount 
of  steel  with  concrete  so  as  to  give  to  this  material  a  reliable 
tensile  and  bending  resistance  has  opened  the  way  for  its  use 
in  a  great  variety  of  forms,  not  only  as  complete  structures, 
or  important  members  of  structures,  but  also  in  many  special 
individual  forms.     Concrete  piles  are  valuable  substitutes  for 
piles  of  wood  where  the  latter  would  be  subject  to  deteriora- 
tion.    Reinforced-concrete  ties  offer  some  evident  advantages 
over  ties  of  wood  or  steel.    This  material  is  also  well  adapted 
to  many  other  special  uses,  particularly  where  durability  is 
.an  important  factor. 


CHAPTER  II. 

PROPERTIES   OF   THE   MATERIALS. 

12.  In  a  design  where  two  or  more  materials  are  combined 
in  the  same  member  the  stresses  in  the  different  materials 
depend  upon  the  elastic  properties  as  well  as  upon  the  super- 
imposed loads.     Therefore  in  making  such  designs  a  knowl- 
edge of  these  elastic  properties  is  quite  as  necessary  as  a  knowl- 
edge of  the  strength  of  the  materials. 

CONCRETE. 

13.  General  Requirements. — The  conditions  to  be  met  in 
reinforced-concrete    construction    require    the    use,    generally, 
of  a  concrete  of  relatively  high  grade.     In  this  type  of  con- 
struction the  strength  of  the  material  is  of  much  greater  im- 
portance than  it  is  in  many  forms  of  plain  concrete  design, 
as  the  dimensions  of  the  structures  are  more  directly  dependent 
upon  strength  and  less  upon  weight.     A  comparatively  strong 
concrete  is  therefore  found  to  be  economical. 

It  is  especially  important,  also,  that  the  concrete  be  of 
uniform  quality  and  free  from  voids,  as  the  sections  are  com- 
paratively small  and  the  stability  of  the  structure,  to  a  much 
greater  extent  than  is  the  case  with  massive  concrete,  is  de- 
pendent upon  the  integrity  of  every  part.  Thoroughly  sound 
concrete  is  also  required  in  order  to  insure  good  adhesion  to 
the  steel  reinforcement  and  adequate  protection  of  the  steel 
from  corrosion  and  from  fire.  These  requirements  call  for 
great  care  in  the  preparation  and  placing  of  the  material. 

8 


§  16.]  PROPERTIES  OF  CONCRETE.  9 

Concrete  is  subject  to  great  variations  in  its  properties, 
owing  to  the  great  variations  in  the  character  and  proportions 
of  its  ingredients  and  in  its  preparation.  It  is  therefore  diffi- 
cult to  judge  from  results  of  tests  made  under  certain  con- 
ditions as  to  what  may  fairly  be  expected  of  a  concrete  pre- 
pared under  other  conditions;  so  that  it  is  very  important 
that  regular  and  systematic  tests  of  the  material  as  actually 
used  be  made  during  the  progress  of  the  work. 

14.  Cement. — Portland   cement   only  should  be  used;    it 
should  meet  such  standard  specifications  as  those  of  the  Ameri- 
can Society  of  Civil  Engineers.     The  rapidity  of  hardening  of 
different  cements  varies   considerably  and  may  be  an  element 
requiring  special  attention  where  the  structure  is  to   receive 
its  load  very  early  or  where  such  load  is  to  be  long  deferred. 

15.  Sand. — The  sand  should  be  free  from  clay  and  pref- 
erably of  coarse  grain.      A  fine  sand    requires  more   cement 
than  a  coarse  sand  for  equal  strength,  and  more  water  for  a 
like  consistency.      In  the  case  of  a  very  fine  sand  the  differ- 
ence may  be  very  marked,  so  that  unless  care  is  taken  and 
special  tests  made,  the  resulting  concrete  is  likely  to  be  porous 
and  deficient  in  strength  and  adhesive  power.     Where  the  use 
of  fine  sand  is  contemplated,  tests  of  strength  may  show  that 
a  considerable  extra  cost  may  be  justified  in  securing  a  coarser 
material.     The  effect  of  size  of  sand  is  shown  in  Art.  19. 

1 6.  Broken  Stone  and  Gravel. — Both  materials  are  satis- 
factory, but  they  should  be  screened  to  remove  the  dust  or 
sand  and  to  remove  particles  larger  than  the  maximum  size 
desired.     Beyond  this,  the  screening  of  stone  to  size  is  unde- 
sirable unless  an  artificial  mixture  is  to  be  made,  as  it  tends  to 
increase  the  proportion  of  voids.     Gravel  may  be  sufficiently 
uniform  in  quality  so  that  the  sand  need  not  be  removed,  but 
it  will  usually  require  screening  in  order  to  insure  a  concrete 
of  definite  proportions. 

,„  The  maximum  desirable  size  of  stone  or  gravel  depends  upon 
the  size  of  the  structural  forms  and  the  size  and  spacing  of  the 
reinforcement,  it  being  desirable  to  use  as  large  a  size  of  aggre- 


10  PROPERTIES  OF   THE  MATERIALS.  [Cn.  II. 

gate  as  will  admit  of  convenient  working.  Maximum  sizes  of 
stone  of  f  inch  to  1|  inches  are  common,  but  on  heavy  work, 
with  rods  widely  spaced,  there  is  no  objection  to  still  larger 
sizes. 

The  crushing  strength  of  a  gravel  concrete  is  usually  a 
little  less  than  one  of  broken  stone  of  the  same  proportion  of 
voids,  but  the  difference  is  unimportant.  The  difference  in 
tensile  strength  is  not  well  determined,  but  the  few  tests  avail- 
able indicate  about  the  same  relative  difference  as  in  eom- 
pressive  strength. 

17.  Proportions  of  Ingredients.; — The  proportions  commonly 
used  vary  from  about  1:1J:3  to  1:3:6  of  cement,  sand,  and 
broken  stone   respectively;    or  the  equivalent  proportions   if 
gravel  be  used.     Richer  mixtures  than  1:2:4  are  not  common, 
nor  poorer  mixtures  than  1:2J:5,  although  with  well-graded 
material  a  very  satisfactory  concrete  can  be  made  of  1:3:6  pro- 
portions.     Occasionally  where   the  design  is   determined   by 
other  considerations  than  strength,  and  cost,  a  very  rich  mix- 
ture or  a  poor  one  may  be  desirable,  but  where  these  elements 
determine  the  design,  the  most  economical  concrete  will  be 
a  rich   concrete   of   about   the   proportions   above   indicated. 
Customary  proportions,  such  as  1:2:4,  should  not  be  blindly 
adopted.    In  any  important  work  a  careful  study  of  the  mate- 
rials and  of  the  best  proportions  to  use  for  economy  and  strength 
will  be  well  repaid.     To  secure  sound  and  reliable  work,  with 
good  adhesion  and  tensile  strength,  there  must  be  no  unfilled 
voids  in  the  stone  and  little  or  none  in  the  sand.     The  former 
is  of  more  importance  than  the  latter,  and  if  cost  and  strength 
are  to  be  reduced  it  should  be  done  by  using  a  poorer  mortar 
to  fill  the  voids  in  the  stone.     For  equal  amounts  of  cement, 
the  denser  the  mixture  (or  the  smaller  the  percentage  of  voids) 
the  stronger  the  concrete.* 

1 8.  Consistency. — The   tendency  in  all  kinds  of  concrete 
construction  is  to  use  a  wetter  mixture  than  formerly.     Rela- 
tively dry  concrete  thoroughly  tamped  will  give  slightly  greater 

*  See  valuable  paper  by  Fuller  and  Thompson  on  "Proportioning  of 
Concrete,"  Trans.  Am.  Soc.  C.  E.,  1907,  LIX,  p.  67. 


§19.]  PROPERTIES    OF    CONCRETE.  11 

strength  than  a  wet  mixture;  however,  if  not  too  wet  the 
difference  is  not  great,  and  considering  the  difficulty  and  ex- 
pense of  securing  the  necessary  amount  of  tamping  of  the 
dry  mixture,  better  results  can  usually  be  secured  by  using  a 
plastic  mixture.  This  is  especially  true  with  reference  to 
obtaining  a  dense,  homogeneous  concrete.  The  usual  prac- 
tice now  is  to  make  the  consistency  such  that  the  concrete  will 
require  only  moderate  tamping  or  puddling  to  bring  the  mass 
to  a  homogeneous  condition.  Such  concrete,  while  somewhat 
weaker  than  the  ideal  compacted  concrete,  will,  under  actual 
conditions,  be  much  more  reliable  and  will  be  free  from  voids. 
In  the  case  of  reinforced-concrete  work  reliability  is  more 
important  than  maximum  strength,  and  is  promoted  by  using 
concrete  of  such  consistency  that  it  can  readily  be  worked 
into  place  in  the  forms  and  around  the  reinforcing  steel.  In 
practice  the  consistency  'varies.  Some  use  a  concrete  which 
requires  considerable  tamping  and  working,  while  others  use 
a  concrete  which  will  practically  flow  into  place.  The  dryer 
the  concrete  the  closer  the  inspection  required  when  the  mate- 
rial is  placed;  on  the  other  hand  very  wet  concrete  is  not  as 
strong  and  needs  to  be  promptly  poured  to  prevent  segrega- 
tion of  the  materials. 

19.  Compressive  Strength. — The  compressive  strength  of 
concrete  is  dependent  upon  many  factors  so  that  it  is  diffi- 
cult and  at  the  same  time  somewhat  misleading  to  present 
" average  values".  Obviously,  in  any  important  work,  the 
strength  should  be  determined  under  the  actual  conditions 
under  \vhich  the  concrete  is  used.  Uniformity  is  quite  as 
important  as  average  strength. 

One  of  the  best  series  of  tests  is  that  made  at  the  Water- 
town  Arsenal  for  Mr.  George  A.  Kimball,  Chief  Engineer  of 
the  Boston  Elevated  Railway  Company.*  The  concrete  was 
made  of  five  brands  of  Portland  cement,  coarse,  sharp  sand, 
and  broken  stone  up  to  2 1 -inch  size.  The  concrete  was  well 
rammed  into  the  molds,  water  barely  flushing  to  the  surface. 

*  Tests  of  Metals,  1869,  p.  717. 


12 


PROPERTIES  OF  THE  MATERIALS. 


[  H.  II. 


The  specimens  were  buried  in  wet  ground  after  being  taken 
from  the  molds.    The  average  results  were  as  follows: 

TABLE  No.  1. 

COMPRESSIVE  STRENGTH  OF  CONCRETE. 

WATER-TOWN  ARSENAL,  1899. 


Mixture. 

Brand  of  Cement. 

Strength,  Pounds  per  Square  Inch. 

7  Days. 

1  Month. 

3  Months. 

6  Months. 

1:2:4  • 

f 
1:3:6  < 

Saylor 

1724 

1387 
904 
2219 
1592 

2238 
2428 
2420 
2642 
2269 

2702 
2966 
3123 
3082 
2608 

3510 
3953 
4411 
3643 
3612 

Atlas  

Alpha  

Germania  

Alsen 

Average    .... 

1565 

1625 
1050 
892 
1550 
1438 

2399 

2568 
1816 
2150 
2174 
2114 

2896 

2882 
2538 
2355 
2486 
2349 

3826 

3567 
3170 
2750 
2930 
3026 

Saylor  

Atlas  

Alpha 

Germania 

Alsen.  
Average  

1311 

2164 

2522 

3088 

In  a  series  of  tests  made  at  the  Watertown  Arsenal  for 
Mr.  George  W.  Rafter,  the  following  average  values  were 
obtained  on  concrete  about  20  months  old.*  The  voids  in 
the  broken  stone  were  practically  filled.  The  mixture  was  of 
damp-earth  consistency : 

Cement.  Sand.  Strength. 

1  1  4467  Ibs/in2 

1  2  3731       " 

1  3  2553       " 

Results  as  high  as  indicated  by  the  preceding  values  can- 
not be  safely  counted  upon  in  practice.  Wet  concrete  will  show 
a  lower  strength  than  concrete  as  dry  as  that  in  the  above 
tests,  especially  for  the  earlier  periods,  but  the  difference 
becomes  less  with  lapse  of  time,  and  a  fairly  soft  plastic  con- 


*  Tests  of  Metals,  1898. 


§  19.]  PROPERTIES  OF  CONCRETE.  13 

crete  will  acquire  about  the  same  strength  as  dry  concrete 
within  three  or  four  months.  A  very  wet  concrete  will,  how- 
ever, continue  to  be  somewhat  weaker  than  one  containing 
less  water,  and  while  such  a  concrete  may,  on  the  whole,  be 
desirable,  its  deficiency  in  strength  as  compared  to  maximum 
values  should  not  be  overlooked.  Other  variations  in  condi- 
tions, such  as  rapid  drying  out,  or  the  use  of  very  fine  sand, 
for  example,  may  give  results  materially  below  those  here 
quoted. 

The  following  average  results  of  a  large  number  of  tests 
in  the  series  made  for  Mr.  Rafter,  already  referred  to,  show 
the  relative  strengths  of  diy,  plastic,  and  wet  concrete  at  the 
age  of  about  twenty  months.  The  dry  mixtures  were  only  a 
little  more  moist  than  damp  earth  and  required  much  ramming; 
the  plastic  mixtures  required  a  moderate  amount  of  ramming 
to  bring  water  to  the  surface;  the  wet  mixtures  quaked  like 
liver  under  moderate  ramming.  Five  brands  of  cement  were 
used  : 

Consistency.  Mean  Compressive  Strength. 

Dry 2348  lbs/in2 

Plastic 2203       " 

Wet..... 2129       " 

In  actual  practice  results  are  very  likely  to  be  less  favorable 
to  dry  mixtures  on  account  of  the  great  difficulty  of  securing 
adequate  tamping. 

The  effect  of  size  of  sand  has  been  thoroughly  investigated 
by  Feret.  Fig.  1,  from  Johnson's  "Materials  of  Construction", 
shows  results  obtained  by  Feret  on  1 : 3  mortar  after  hardening 
one  year  in  fresh  water.  The  sand  used  consisted  of  mixtures  of 
various  proportions  of  fine  (.0  to  .5  mm.),  medium  (.5  to  2  mm.), 
and  coarse  (2  to  5  mm.)  sand,  and  in  the  figure  the  result  from 
any  particular  mortar  is  recorded  in  the  triangle  at  such  dis- 
tances from  the  three  base-lines  as  will  represent  the  propor- 
tions of  each  size  sand  used.  Lines  of  equal  strength  were 
then  drawn  in  the  diagram.  Thus  the  strength  of  the  mortar 


14  PROPERTIES    OF    THE   MATERIALS.  [Cn.  II. 

in  which  only  fine  sand  was  used  was  only  1400  lbs/in2.  The 
maximum  strength  of  3500  lbs/in2  was  obtained  from  a  mix- 
ture containing  about  85%  of  coarse  sand  and  15%  of  fine, 
with  a  very  little  sand  of  medium  size.  This  diagram  shows 
in  a  striking  manner  the  effect  of  size  of  sand. 


FIG.  1.— Effect  of  Size  of  Sand. 

The  tests  quoted  in  the  foregoing  were  made  on  specimens 
of  cube  form,  which  has  generally  been  considered  the  standard 
form  of  compression  specimen.  More  recently,  however,  the 
prismatic  form,  of  a  height  of  2  to  3  diameters,  has  been  com- 
monly used  because  of  its  advantages  in  the  measurement  of 
distortions  and  in  studying  the  results  of  tests  on  columns. 
In  the  prismatic  form  the  strength  will  generally  be  from  10 
to  30%  less  than  in  the  cube  form,  as  there  is  greater  freedom 
for  shearing  action  to  take  place.  The  results  are,  however, 
likely  to  be  more  uniform  as  they  depend  less  upon  the  nature 
of  the  bedding,  etc.  For  comparison  with  the  strength  of 
concrete  in  a  beam  the  cube  form  is  satisfactory. 

The  effect  of  varying  conditions  on  the  quality  of  concrete 
is  well  shown  by  results  obtained  at  various  times  in  the  labor- 
atories of  the  University  of  Wisconsin.*  Tests  made  in  1906 


*  See  Bulletins  No.  1  and  2,  Vol.  4,  Engineering  Series. 


§20.]  PROPERTIES  OF  CONCRETE.  15 

on  thirty-six  cylinders,  6X18  in.  in  size,  of  1:2:4  concrete, 
30  days  old,  hand-mixed,  gave  an  average  strength  of  1750 
lbs/in2,  with  many  results  below  1500.  Tests  in  1907  on 
twenty-five  cylinders,  10X24  in.  in  size,  of  machine-mixed 
concrete  of  the  same  kind,  gave  an  average  strength  of  1940 
lbs/in2,  individual  results  varying  from  1500  to  2600  lbs/in2. 
A  fairly  fine  sand  was  used  and  the  specimens  were  cured  in 
air.  More  recent  tests  indicate  that  with  good  materials  and 
careful  manipulation  an  average  strength  of  2200  lbs/in2  can 
readily  be  obtained  on  cylinders  for  60-day  tests. 

Considering  the  various  results  noted  it  may  be  concluded 
that  under  reasonably  good  conditions  as  to  character  of  ma- 
terial and  workmanship  an  average  strength  of  about  2000 
lbs/in2  may  be  expected  of  a  1:2.4  concrete  in  30  to  60  days, 
on  cylindrical  specimens,  the  rate  of  hardening  depending  upon 
the  consistency  and  the  temperature,  and  for  a  1.3:6  concrete 
a  strength  of  about  1600  lbs/in2. 

It  is  important  that  the  strength  be  determined  by  actual 
tests  of  the  material  proposed  to  be  used,  and  if  the  results  are 
too  low  the  ingredients  or  proportions  should  be  modified  until  a 
satisfactory  result  is  obtained.  Where  the  usual  proportions 
give  low  results  it  will  generally  be  advisable  to  increase  the 
richness  of  the  concrete  rather  than  to  reduce  the  working 
stresses. 

20.  Tensile  Strength. — The  tensile  strength  of  concrete  is 
quite  as  important  as  the  compressive  strength.  In  fact  the 
most  common  type  of  failure  of  a  reinforced  concrete  beam  is 
closely  related  to  the  tensile  strength  of  the  concrete.  The 
tensile  strength  is  generally  from  one-tenth  to  one-twelfth  of 
the  compressive  strength,  but  this  ratio  varies  considerably. 
The  character  of  the  material  and  workmanship  has  probably 
a  greater  influence  upon  the  tensile  strength  than  Upon  the 
compressive. 

Tests  by  Mr.  M.  0.  Withey  on  1:2:4  concrete,  28  days 
old,  gave  results  averaging  189  lbs/in2,  varying  from  142 
to  160  lbs/in2.  The  compressive  strength  was  1940 


16  PROPERTIES   OF    THE    MATERIALS.  [Cn.  II. 

lbs/in2.*     Tests  by  Mr.  W.  H.  Henby  f  gave  results  as  fol- 
lows:   . 

Mixture.          Compressive  Strength.  Tensile  Strength. 

1:2:4  3000  lbs/in2  180  lbs/in2 

1:3:6  1800       "  115       " 

Tests  by  Professor  W.  K.  Hatt  f  gave  the  following  results: 

„.    ,      .  ~  Age.     Compressive  Strength,    Tensile  Strength, 

Kind  of  Concrete.  *  Ibg/in2  Ibg/in2 


1:2:4  (broken  stone)  30  311 

1:2:5  "  90  2413  359 

1:2:5  "  28  2290  237 

1:5  (gravel)  90  2804  290 

1:5        "  28  2400  253 

Tests  by  Professor  Ira  H.  Woolsen§  on  1:2:4  mixtures 
5  to  7  weeks  old  gave  an  average  tensile  strength  of  161  lbs/in2, 
compared  to  1753  lbs/in2  Compressive  strength. 

Professor  Talbot  obtained  values  for  1:3:6  concrete  from 
50  to  84  days  old  of  178,  160,  and  170  Ibs/in2.!l 

21.  Tensile  Strength  as  Determined  by  Transverse  Tests.— 
The  transverse  strength  of  plain  concrete  depends  almost 
entirely  upon  its  tensile  strength,  although  the  modulus  of 
rupture  is  considerably  greater  than  the  strength  in  plain  ten- 
sion owing  to  the  curved  form  of  the  stress-strain  diagram. 
Feret^  found  a  very  nearly  constant  ratio  of  1.95  of  modulus 
of  rupture  to  tensile  strength.  The  value  of  this  ratio  will 
ordinarily  range  from  1.8  to  2.  Transverse  tests  of  different 
concretes  should  therefore  show  about  the  same  relative  results 
as  tensile  tests.  They  are  in  fact  quite  as  significant  in  this 
connection. 

Some  of  the  best  tests  on  transverse  strength  are  those 
made  by  William  B.  Fuller,  and  given  in  full  in  Taylor  and 

*  Bulletin  No.  2,  Vol.  4,  Fniv.  of  Wis.,  1908.       ~ 

t  Jour.  Assn.  Eng.  Soc.,  Sept.  1900. 

t  Jour.  West.  Soc.  Eng.,  Vol.  IX,  1904,  p.  234. 

§  Eng.  News,  Vol.  LIII,  1905,  p.  561. 

II  Bulletin  No.  1,  Univ.  of  111.,  1904. 

Tf  Etude  Experimentale  du  Ciment  Arme.      Paris,  1906. 


§22.]  PROPERTIES  OF  CONCRETE.  17 

Thompson's  work  on  Concrete.*    The  following  average  results 
were  obtained  for  33-35-day  tests. 

Mixture  by  Volume. 


1:2.16:4.08  439  lbs/in2 

1:2.16:5.1  380       il 

1:3.24:5.1  285       " 

1:3.24:6.12  226       " 

1:3.24:7.14  239       " 

Here  we  find  the  strength  of  the  1:3.24:6.12  mixture  only 
about  one-half  that  of  the  1:2.16:4.08  mixture,  indicating  the 
relative  weakness  in  tension  of  the  lean  mixture. 

The  results  herein  given,  both  of  tensile  and  of  transverse 
tests,  indicate  that  the  quality  of  the  concrete  has  a  greater 
relative  effect  on  the  tensile  strength  than  on  the  compressive 
strength,  the  strength  of  a  1:3:6  mixture  being  not  more  than 
twro-thirds  that  of  a  1:2:4  mixture.  Reasonable  values  for 
ultimate  tensile  strength  would  appear  to  be  about  as  follows: 

1:2:4  mixture  .........   160-200  lbs/in2 

1:3:6       "       .........    100-125      " 

22.  Shearing  Strength.—  There  is  a  lack  of  uniformity  among 
writers  as  to  just  what  is  meant  by  the  term  "shearing  strength  ", 
resulting  in  a  wide  variation  in  the  suggested  values  for  working 
stresses.  In  this  work  the  authors  will  use  the  term  as  it  is 
commonly  thought  of  among  American  engineers,  to  denote 
the  strength  of  the  material  against  a  sliding  failure  when  tested 
as  a  rivet  or  bolt  would  be  tested  for  shear;  that  is,  when  the 
maximum  shearing  stresses  are  confined  to  a  single  plane. 

Tests  made  under  the  direction  of  Professor  C.  M.  Spofford 
on  cylinders  5  inches  in  diameter  with  ends  securely  clamped 
in  cylindrical  bearings  gave  results  as  follows: 


Shearing 

Compressive 

Ratio  of 

Mixture. 

Strength, 

Strength, 

Shearing  to 

lbs/in2, 

lbs/in2. 

Comp.  Strength. 

1:2:4 

1480 

2350 

.63 

1:3:5 

1180 

1330 

.89 

1:3:6 

1150 

1110 

1.04 

*  Concrete,  Plain  and  Reinforced.     N.  Y.,  1906. 


18  PROPERTIES   OF  THE  MATERIALS.  [Cn.  II. 

Tests  made  at  the  University  of  Illinois  on  rectangular  speci- 
mens tested  in  a  similar  manner  gave  the  following  average 
results : 

Shearing  Compressive  Ratio  of 

Mixture.  Strength,  Strength,  Shearing  to 

Ibs/in2.  lbs/in2.          Comp.  Strength. 

1:2:4  1418  3210  .44 

1:3:6  1250  2290  .57 

Tests  made  by  punching  through  plates  gave  shearing 
strengths  varying  from  37  to  90  per  cent  of  the  compressive, 
the  value  depending  upon  the  form  of  test-piece.* 

Tests  by  M.  Feret  on  mortar  prisms  gave  results  for  shear- 
ing strength  equal  to  about  one-half  the  crushing  strength. 

The  ordinary  crushing  failure  is  really  a  failure  by  shear- 
ing, and  under  such  conditions  the  crushing  stress  is,  theo- 
retically, twice  the  .shearing  stress,  the  angle  of  shear  being 
45°.  Results  of  tests  give  a  somewhat  greater  inclination 
than  45°,  so  that  the  crushing  stress  is  somewhat  greater  than 
twice  the  actual  shearing  stress. 

We  may  then  conclude,  both  from  theory  and  from  tests, 
that  the  shearing  strength  of  concrete,  in  the  sense  here  used, 
is  nearly  one-half  the  crushing  strength.  It  is  in  fact  so  large 
that  it  will  need  to  be  considered  only  in  exceptional  cases. 

Some  writers  used  the  term  " shearing  stress"  to  mean 
quite  a  different  thing  from  that  discussed  above,  namely, 
the  complex  action  which  occurs  in  the  web  of  a  beam.  In 
this  case  there  exist  direct  tensile  and  compressive  stresses 
which  at  the  neutral  axis  are  equal  in  intensity  to  the  vertical 
and  horizontal  shearing  stresses.  The  limit  of  distortion  in 
the  concrete  will  be  reached,  and  failure  will  occur,  when  the 
tensile  strength  of  the  material  is  exceeded.  Such  a  failure 
may  perhaps  be  called  a  shearing  failure,  but  is  more  strictly 
a  failure  in  tension  in  a  diagonal  direction,  and  is  so  considered 
in  this  work.  Treated  as  a  shearing  failure  the  strength  should 
be  very  nearly  the  same  as  the  tensile  strength  of  the  material 
determined  in  the  usual  way.  In  practice  the  diagonal  tensile 

*  Bulletin  No.  8,  Univ.  of  111.,  1906. 


§23.]  PROPERTIES  OF  CONCRETE.  19 

stresses  in  a  beam  must  often  be  considered,  but  shearing 
stresses,  as  such,  will  be  dangerous  only  in  exceptional  circum- 
stances, such  as  exist  where  a  heavy  load  is  applied  close  to 
a  support. 

23.  Elastic  Properties  of  Concrete. — Stress-strain  Curve 
in  Compression. — In  the  design  of  combination  structures,  such 
as  those  of  steel  and  concrete,  it  is  necessary  to  know  the 
relative  stresses  under  like  distortions.  These  will  depend 
upon  the  moduli  of  elasticity  of  the  two  materials.  For  purposes 
of  safe  design  we  need  to  know  also  the  elastic-limit  strength. 

Fig.  2  represents  typical  stress-strain  curves  for  concrete 
in  compression.  Curves  C,  D,  E,  and  F  were  obtained  at  the 
University  of  Wisconsin  from  tests  on  cylinders  6  inches  in 
diameter  by  18  inches  high.  The  concrete  was  1:2:4  limestone 
concrete  30  days  old.  The  ultimate  strengths  ranged  from  1500 
to  2300  lbs/in2.  Curves  A  and  B  are  typical  curves  selected 
from  the  Watertown  Arsenal  tests  already  quoted,  and  repre- 
sent 1:2:4  and  1:3:6  concrete  respectively. 

Unlike  the  elastic  line  for  steel,  the  line  for  concrete  is 
slightly  curved  almost  from  the  beginning,  the  curvature  grad- 
ually increasing  towards  the  end.  There  is,  however,  no  point  of 
sharp  curvature  as  for  ductile  materials.  A  release  of  load  at 
a  moderate  stress,  such  as  500  to  600  lbs/in2,  will  usually  show 
a  small  set  indicating  imperfect  elasticity.  A  second  applica- 
tion of  the  load  will,  however,  give  a  straighter  line  than  the 
first  and  there  will  be  much  less  permanent  set  following  the 
release  of  load.  After  a  few  repetitions  of  load  there  will  be 
no  further  set  and  the  stress-strain  line  will  become  a  straight 
line  up  to  the  load  applied.  There  is  a  limit  of  stress,  however, 
beyond  which  repeated  applications  of  load  will  continue  to 
add  to  the  permanent  deformation  and  the  specimen  will 
ultimately  fail.  The  general  behavior  under  repeated  stress 
is  indicated  in  Fig.  3,  from  tests  on  concrete  similar  to  those 
represented  in  Fig.  2.  For  a  very  exhaustive  study  of  this 
subject  the  reader  is  referred  to  the  work  of  Bach.* 
*  Zeit.  V.  dt.  Ing.,  1895,  etc. 


20 


PROPERTIES  OF  THE  MATERIALS. 


[Cn.  II. 


24.  Modulus    of    Elasticity    in    Compression. — The    stress- 
strain  line  being  curved  almost  from  the  beginning,  the  proper 


2400 
2200 
2000 
1800 
1600 
1400 
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1000 
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Deformation  per  Unit  Length 
FIG.  2. — Compressive  Stress-strain  Diagrams  of  Concrete. 

method   of   calculating  the   modulus  of  elasticity  needs   to  be 
considered.     Fig.  4  is  a  typical  stress-strain  diagram  for  com- 


§24.] 


PROPERTIES  OF  CONCRETE. 


21 


pression  (somewhat  simplified),  B  and  C  being  points  where 
the  loads  have  been  removed  and  reapplied.  For  very  low 
stresses,  up  to  perhaps  300  to  400  lbs/in2  (a  low  working 
stress),  the  variation  of  the  curve  from  a  straight  line  is  so  small 


Load  in  Pounds  per  Square  Inch 

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Deformation  per  Unit  Length 

FIG.  3. — Stress-strain  Diagram  under  Repeated  Loads. 

that  it  may  be  considered  as  straight,  and  an  average  straight  line 
may  be  drawn,  as  OT,  and  its  slope  taken  as  the  modulus  of  elas- 
ticity. This  line  may  be  considered  the  same  as  the  tangent 
at  the  origin.  For  higher  stresses,  reaching  to  a  point  along 


22 


PROPERTIES  OF  THE  MATERIALS. 


[Cn.  II. 


the  curved  portion  such  as  point  B,  it  is  usual  to  deduct  the 
permanent  set  Oa  from  the  deformation  Ob  and  divide  the  stress 
by  the  remaining  elastic  deformation  ab.  This  gives  the  slope 
of  the  line  aB,  and  may  be  considered  to  represent  the  law  of 
elastic  deformation  for  stresses  within  the  limit  of  the  stress  bB 
after  the  first  few  applications  of  load.  A  modified  " elastic" 
curve,  OB'O ',  can  thus  be  drawn  by  deducting  from  the  defor- 
mation for  each  load  the  subsequent  set,  giving  a  steeper  curve 
and  one  more  nearly  approaching  a  straight  line.  On  the  basis 


b  Deformation 
FIG.  4. 

of  this  "elastic"  curve  the  modulus  of  elasticity  for  stresses 
up  to  any  given  maximum  would  then  be  equal  to  that 
maximum  stress  divided  by  the  elastic  deformation  at  that 
stress. 

There  being  no  general  agreement  as  to  the  exact  definition  of 
the  word  " modulus"  for  such  materials  as  concrete,  the  method 
which  should  be  employed  in  calculating  its  value  should  depend 
upon  the  purpose  for  which  it  is  to  be  used.  The  principal 
use  of  the  modulus  of  elasticity  in  reinforced- concrete  design 
is  to  determine  the  relative  stresses  carried  by  the  concrete 
and  the  steel  in  compression  members,  and  to  find  the  neutral 
axis  in  beams.  After  the  neutral  axis  is  once  found  the  modulus 
does  not  enter  into  the  calculations. 


§24.]  PROPERTIES  OF  CONCRETE.  23 

Consider  the  action  in  the  case  of  a  column.  Assuming 
no  initial  stress  in  the  steel  or  concrete,  suppose  that  the 
column  is  loaded  so  as  to  cause  a  shortening  equal  to  Ob,  Fig. 
4.  The  stress  in  the  concrete  will  be  bB,  and  that  in  the  steel 
will  be  equal  to  the  deformation  06  multiplied  by  its  modulus 
of  elasticity.  Upon  removal  of  the  load  there  may  be  a  per- 
manent set  Oa,  which  means  that  there  is  some  residual  com- 
pression in  the  steel  (with  an  equal  amount  of  tension  in  the 
concrete).  A  second  application  of  the  load  will  cause  a 
d .formation  ab,  but,  measuring  from  the  original  position,  the 
deformation  is  Ob,  and  this  again  fixes  the  stress  in  the  steel. 
Hence,  for  the  determination  of  the  relative  stresses  in  steel  and 
concrete,  the  modulus  for  the  concrete  should  be  the  ratio  of 
Bb  to  Ob,  or  the  slope  of  the  chord  OB. 

In  the  case  of  a  beam  the  stresses  in  the  concrete  at  any 
section  will  vary  from  zero  at  the  neutral  axis  to  the  value 
Bb,  for  example,  at  the  extreme  fibre.  At  intermediate  points 
the  stresses  follow'  approximately  the  law  of  the  curve  OB. 
In  this  case  a  chord  OB  does  not  exactly  represent  the  facts, 
but  the  error  is  small,  and  it  is  the  best  line  to  use  if  the  rec- 
tilinear variation  of  stress  be  assumed.  If  a  curvilinear  law 
is  used,  then  the  modulus  is  supposed  to  be  the  slope  of  the 
tangent  at  the  origin.  In  neither  case  is  it  correct  to  use  the 
slope  of  the  line  aB. 

In  referring  to  these  various  methods  of  calculating  the 
modulus,  the  slope  of  the  tangent  OT  is  generally  called  the 
"initial  modulus."  The  slope  of  the  line  OB  may  be  called 
the  "secant  modulus." 

The  value  of  the  modulus  for  concrete  varies  greatly  as 
determined  by  different  experimenters  and  for  different  kinds 
of  concrete.  As  a  rule  the  denser  and  older  the  concrete  the 
higher  the  modulus. 

Among  the  most  careful  experiments  are  those  by  Bach,* 
in  which  he  repeated  the  loads  at  each  increment  until  there 
was  practically  no  increase  of  set. 

*  Zeit.  V.  dt.  Ing.,  1895. 


24  PROPERTIES  OF  THE  MATERIALS. 

The  following  are  some  average  results: 


[Cn.  II. 


Kind  of  Concrete. 

Modulus  of  Elasticity,  lbs/in2. 

Based  on  Elastic  Deformation. 

Based  on  Total 
Deformation. 

At  114  lbs/in2. 

At  570  lbs/in2. 

3,590,000 
2,520,000 
2,990,000 
2,240,000 

At  570  lbs/in2. 

1:2^:5  (broken  stone)  
1  -9A-5  travel) 

4,660,000 
3,170,000 
3,870,000 
3,000,000 

3,440,000 
2,2CO,000 
2,570,000 
2,110,000 

1  '3'6  (broken  stone)  

1  •  3  •  6  (gravel)      

The  specimens  were  25  cm.  in  diameter  and  100  cm.  high 
and  were  from  three  to  four  months  old. 

The  average  values  of  the  modulus  obtained  in  the  Water- 
town  Arsenal  tests  mentioned  in  Art.  19  were  as  follows: 

TABLE  No.  2. 

MODULUS    OF   ELASTICITY    OF   CONCRETE. 
WATERTOWN  ARSENAL  TESTS,   1899. 


Mixture. 

f 

1:2:4  \ 

i 

f 

1:3:6  •{ 

i 
i 
<. 

Brand  of  Cement. 

Modulus  of  Elasticity  between  Loads  of  100 
and  600  lbs/in2,  Based  on  Elastic 
Deformation. 

7-10  Days. 

1  Month. 

3  Months. 

Savior 

1,667,000 
2,778,000 
1,000,000 
2,500,000 
2,500,000 

2,500,000 
3,125,000 
2,083,000 

2,778,000 

3,571,000 
4,167,000 
4,167,000 
3,571,000 
2,778,000 

Atlas 

Alpha   

Germania  

Alsen  

Average  .  . 

2,089,000 

2,273,000 
1,667,000 

2,273,000 
1,667,000 

2,621,000 

2,778,000 
3,125,000 
2,083,000 
2,273,000 
2,273,000 

3,651,000 

4,167,000 

2,778,000 
3,571,000 
2,778,000 
2,778,000 

Say  lor.  ... 

Atlas  

Alpha 

Germania. 

Alsen  

Average 

1,970,000 

2,506,000 

3,214,000 

These  results  were  calculated  by  using  the  total  deformation 
minus  the  set.  If  the  total  deformation  be  used  the  values 
would  be  reduced  in  most  cases  10  to  20  per  cent. 


§25.]          PROPERTIES  OF  CONCRETE.  25 

Tests  made  at  the  University  of  Wisconsin  show  a  large 
variation  of  the  modulus  with  variation  in  the  quality  of  the 
concrete.  Tests  in  1906  on  30  prisms  of  1:2:4  concrete,  30 
days  old,  gave  an  average  value  of  2,560,000  lbs/in2  at  a  stress 
of  600  lbs/in2,  using  total  deformation.  Similar  tests  in  1907 
gave  an  average  value  of  3,500,000  lbs/in2,  varying  from 
2,800,000  to  3,800,000.  The  respective  average  compressive 
strengths  were  1780  and  1940  lbs/in2,  the  latter  concrete  being 
considerably  denser  than  the  former.*  Values  still  higher 
have  been  found  by  some  experimenters,  but  generally  the 
calculations  have  been  made  with  reference  to  the  tangent  at 
the  origin  or  to  the  elastic  deformation. 

Considering  the  various  results  obtained  and  the  significance 
of  total  deformation  it  would  appear  that  for  working  loads 
the  modulus  for  ordinary  concrete  ranges  from  2,500,000  to 
3,500,000  lbs/in2,  depending  upon  the  mixture  and  the  age  of 
the  concrete.  As  will  be  shown  subsequently,  however,  the 
value  selected  should  also  depend  upon  the  purpose  for  which 
it  is  to  be  used,  and  that  for  most  calculations  relating  to  strength 
a  value  of  2,000,000  is  more  satisfactory  than  a  higher  value. 

25.  Elastic  Limit. — As  stated  in  the  preceding  article,  con- 
crete shows  a  permanent  set  under  small  loads  so  that,  in  the 
usual  sense,  the  material  can  hardly  be  said  to  have  an  elastic 
limit.  There  appears  to  be,  however,  a  limit  to  the  stress 
which  can  be  repeated  indefinitely  without  continuing  to  add 
to  the  deformation,  and  this  limit  may  be  taken  as  the  elastic 
limit  for  practical  purposes.  From  experiments  by  Bach  and 
others,  this  limit  seems  to  be  from  one-half  to  two-thirds  the 
ultimate  strength.  In  repeated-load  experiments  on  neat 
cement  and  on  concrete  made  by  Professor  J.  L.  Van  Ornum  | 
it  has  been  shown  that  the  maximum  load  which  may  be 
repeated  an  indefinite  number  of  times  without  rupture  does 
not  much  exceed  50%  of  the  ultimate  strength  (Art.  117). 
These  results  show  a  close  relation  to  those  obtained  by  Bach, 

*  Bulletins  No.  1  and  2,  Vol.  4,  Univ.  of  Wis. 

t  Trans.  Am.  Soc.  C.  E.,  Vol.  LI,  p.  443.     Proc.  Am.  Soc.  C.  E.,  Dec.  1906. 


26  PROPERTIES    OF    THE   MATERIALS.  [Cn.  II. 

and  it  may  therefore  be  concluded  that  the  limit  of  permanent 
elasticity  for  repeated  loads  is  from  50  to  60%  of  the  ultimate 
strength. 

26.  Comparison  of  Stress-strain  Curve  with  the  Parabola. — 
As    the    parabola    is    often    used    in    theoretical    analyses  to 
represent  the  stress-strain  curve  it  will  be  useful  to  compare 
some  typical  curves  with  the  parabola.    The  form  of  parabola 
used  has  its  axis  vertical  and  its  vertex  at  the  point  of  the 
curve  representing  the  ultimate  strength.     In  Fig.  5  the  curves 
shown  in  Fig.  2  are  compared  with  parabolas  (shown  in  dotted 
lines).    In  the  case  of  curves  C,  D,  E,  and  F  the  agreement  is 
very  close. 

27.  Stress-strain    Curve  for    Tension. — Comparatively    few 
tests  have  been  made  on  the  elasticity  of  concrete  in  tension. 
Bach  found  for  1 . 4  concrete  an  average  value  of  the  modulus 
of  3,800,000  lbs/in2  at  a  stress  of  80  lbs/in2,  and  3,100,000 
at  a  stress  of  135  lbs/in2.    The  ultimate  tensile  strength  was 
185  lbs/in2.     The  modulus  in  compression  for  the  same  con- 
crete was  3,850,000  at  80  lbs/in2.*    Professor  Hatt  has  deter- 
mined  values   ranging   from   2,000,000   to   5,000,000   lbs/in2, 
which  were  generally  about  equal  to  the  values  in  compression. f 
These  and  other  tests  indicate  that  the  initial  moduli  in  ten- 
sion and  in  compression  are  about  the  same,  and  as  the  working 
limit  in  tension  is  very  low  they  may  be  assumed  as  equal. 
The  relative  strength  and  deformation  of  concrete  in   com- 
pression and  tension  is  illustrated  by  a  typical  curve  in  Fig.  6. 

28.  Coefficient  of  Expansion. — Experiments  by  Professor 
W.  D.  Pence  J  on  1-2.4  concrete  gave  an  average  value  of  the 
coefficient   of  expansion  of  .0000055  per  degree  Fahrenheit, 
there  being  little  variation  among  the  several  tests.    Tests 
made  at  Columbia  University  on  1.3:6  concrete  gave  values 
of  about  .0000065.     Other  experiments  have  shown  somewhat 
higher  results.     A  value  of  .000006  may  be  assumed. 

*  Mitt.  Uebcr  Forsch.  a.  d.  Gebiet  des.  Ing.,  1907,  Heft  45-47. 

t  Proc.  Am.  Soc.  Test.  Mat.,  1902. 

J  Jour.  West  Soc.  Eng.,  Vol.  VI,  1901,  p  549. 


§29.] 


PROPERTIES  OF  CONCRETE. 


27 


29.  Contraction  and  Expansion  in  Hardening. — Many  ex- 
periments have  been  made  relative  to  shrinkage  and  swelling 


2400 


2200 


2000 


1800 


1GDO 


1400 


.1200 


1000 


800 


600 


400 


200 
100 
2400 


2200 


2000 


1800 


s 


* 


1600 


1400 
1200 


1000 


800 


600 


400 


200 


t 


.0002  .0004  .0 


.0010   0 


.0002  .0004  .0006  .OOQ8  .0010  .0012  .0014  .0016 


.0002  .0004  .0006  .0008  .0010     0    .0002  .0004  .0006  .0008  .0010  .OQ12 .0014  . 
Deformation  per  Unit  Length 

FIG.  5. — Stress-strain  Curves  Compared  with  Parabolas, 
of  cement-mortar  in  hardening.     In  general  the  results  show 
that  when  hardened  in  air  there  will  be  more  or  less  shrinkage, 


28 


PROPERTIES  OF  THE  MATERIALS. 


[Cn.  II. 


but  when  hardened  in  water  there  is  likely  to  be  some  swelling, 
although  results  on  this  point  are  not  entirely  consistent. 
The  richer  the  mortar  (or  concrete)  the  greater  the  change  in 
dimensions.  Experiments  by  Considere  and  others  indicate 
that  1:3  plain  mortar  will  shrink  .05  to  .15%  when  hardened 


-1000 


.0002   0 


200- 


100- 


0002  .0004 


.0008 


Deformation 


.0012 


FIG.  6. — Relative  strength  and  deformation  in  compression  and  tension. 

in  air  for  2  to  4  months,  and  neat  cement  from  two  to  three 
times  as  much.  Considere  found  the  shrinkage  of  a  1 : 8  mortar 
reinforced  with  5J%  of  steel  to  be  only  .01%,  or  one-fifth  the 
amount  his  tests  showed  on  plain  mortar.  The  few  testa 
available  show  that  the  shrinkage  of  concrete  is  less  than  that 


§31.] 


PROPERTIES  OF  CONCRETE. 


29 


of  mortar,  and  it  would  appear  that  the  shrinkage  should  be 
nearly  proportional  to  the  amount  of  cement  per  unit  volume, 
as  the  sand  and  stone  are  unaffected. 

30.  Weight  of  Concrete. — The  weight  of  concrete  of  the 
usual  proportions  will  vary  from  140  to  150  lbs/ft3,  depending 
upon  the  degree  of  compactness  and  the  specific  gravity  of  the 
materials.    Variation  of  proportions  will  affect  the  weight  but 
little  if  the  proper  ratio  of  sand  and  stone  be  maintained,  but 
a  wet  concrete  when  dried  out  will  weigh  less  than  a  well-com- 
pacted concrete  containing  originally  less  water.     For  prac- 
tical purposes  an  average  value  of  145  lbs/ft3  may  be  taken. 
The  addition  of  reinforcing  steel  in  the  usual  proportions  will 
add  from  3  to  5  pounds,  so  that  the  weight  of  reinforced  con- 
crete may  be  taken  at  150  lbs/ft3. 

31.  Properties  of  Cinder  Concrete. — The  following  table  of 
results  indicates  fairly  well  the  strength  and  modulus  of  elas- 
ticity of  cinder  concrete.    The  age  of  the  specimens  varied 
from  30  to  100  days.    Cinder  concrete  will  weigh  from  110  to 
115  lbs/ft3. 

TABLE  No.  3. 

CRUSHING   STRENGTH    AND    MODULUS    OF   ELASTICITY    OF    CINDER 

CONCRETE. 

WATER-TOWN'    ARSENAL,   TESTS,    1898. 


Mixture. 

Average  Crushing  Strength, 
Ibs  /in  2. 

Average  Modulus  of  Elas- 
ticity between  Loads  of 

100  and  600  Ibs  /in  2. 

Cement. 

Sand. 

Cinders. 

One  Month. 

Three  Months. 

1 

1 

3 

1540 

2050 

2,540,000 

1 

2 

3 

1098 

1634 

1 

2 

4 

904 

1325 

1 

2 

5 

724 

1094 

1,040,000 

1 

3. 

6 

529 

788 

30  PROPERTIES  OF  THE  MATERIALS.  [Cn.  II. 

REINFORCING   STEEL. 

32.  General  Requirements. — In   general,  reinforcing  steel 
must  be  of  such  form  and  size  as  to  be  readily  incorporated 
into  the  concrete  so  as  to  make  a  monolithic  structure.    To 
provide   the  necessary  bond  strength  and   to  distribute   the 
steel  where  needed  without  concentrating  the  stresses  on  the 
concrete  too  greatly,  requires  the  use  of  the  steel  in  compara- 
tively small  sections.    This  requirement,  as  well    as  that  of 
economy  and  convenience,  leads  to  the  use  of  the  steel  in  the 
form  of  rods  or  bars.     These  will  vary  in  size  from  about 
j  to  f  inch  for  light  floors  up  to  1J  to  2  inches  as  maximum 
sizes  for  heavy  beams  or  columns.     Under  certain  conditions 
a  riveted  skeleton  work  is  preferred  for  the  steel  reinforcement, 
but  this  is  usually  where  for  some  reason  it  is  desired  to  have 
the  steelwork  self-supporting  or  where  it  is  to  carry  an  unusu- 
ally large  proportion  of  the  load. 

33.  Forms  of  Bars. — Plain  round  rods  have  been  used  gen- 
erally in  Europe  for  many  years,  and  also  very  largely  in  this 
country,  adhesion  being  depended  upon  for  the  transmission 
of  stress.    Square  rods  show  about  the  same  adhesive  strength 
as  round  rods,  but  are  not  so  convenient  to  use  or  so  readily 
obtained.    Flat  bars  are  undesirable,  as  their  adhesion  to  the 
concrste  is  much  below  that  of  round  or  square  bars. 

Many  special  forms  of  bars  have  been  devised,  the  prin- 
cipal object  of  which  is  to  furnish  a  bond  with  the  concrete 
independent  of  adhesion, — a  " mechanical  bond"  as  it  is  usu- 
ally called.  Some  of  the  most  common  types  of  such  bars 
are  illustrated  in  Fig.  7.  Fig.  (a)  is  the  twisted  square  bar 
invented  by  Mr.  Ransome  and  called  by  his  name.  It  is 
usually  twisted  cold.  Figs.  (6),  (c),  and  (d)  illustrate  various 
well-known  types  of  deformed  bars  which  are  shaped  in  the 
rolling.  Fig.  (e)  illustrates  the  Kahn  bar,  formed  by  turning 
up  strips  sheared  from  the  thin  part  of  the  bar.  Many  other 
devices  are  employed  to  a  greater  or  less  extent  to  provide  a 
mechanical  bond,  and  a  great  variety  of  combinations  of  forms 


§33.] 


PROPERTIES  OF  STEEL. 


31 


FlO.  7. — Deformed  Bars. 


32  PROPERTIES  OF  THE  MATERIALS.  [Cn.  II. 

are  used  in  the  construction  of  beams,  floors,  and  columns  as 
patented  "  systems".  .  It  is  the  purpose  here  to  mention  only 
the  most  common  types  of  bar  element. 

34.  General  Quality  of  the  Steel.—  Steel  used  in  reinforced 
work  is  not  usually  subjected  to  as  severe  treatment  as  that 
used  in  ordinary  structural  work.     Bars  must  be  capable  of 
being  bent  to  the  desired  form,  but  this  is  the  only  treatment 
to  which  the  ordinary  bars  are  subjected.     In  many  concrete 
structures  the  impact  effect  is  also  likely  to  be  less  than  in 
all-steel  structures;    consequently  it  is  considered  that  a  some- 
what less  ductile  material  may  safely  be  used,  but  to  what 
extent  these  considerations  should  permit  the  use  of  steel  of 
cheaper  grade  or  of  higher  elastic  limit  is  an  open  question  on 
which  there  is  much  difference  of  opinion.    The  question  of 
elastic  limit  as  related  to  working  stresses  and  stresses  in  the 
concrete  is  discussed  in  Chapter  V. 

35.  Tensile  Strength.  —  As  regards  tensile  strength  the  steel 
used  is  generally  the  ordinary  mild  or  medium  steel,  or  is  a 
special  high  elastic  limit  material.     The  latter  is  seldom  em- 
ployed where  plain  bars  are  used.     The  ultimate  strength  and 
elastic  limit  of  these  two  general  grades  of  material  are  about 
as  follows: 


Elastic  Limit.  Ultimate  Strength. 

Medium  steel  ......  ____  35-40,000  lbs/in2    60-70,000  lbs/in2      22-25% 

High  elastic  limit  steel.  .  50-60,000      "        80-100,000    "          12-18% 

In  some  forms  of  rods  used  the  elastic  limit  is  artificially 
raised  by  cold  working. 

36.  Modulus  of  Elasticity.—  The  modulus  of  elasticity  of  all 
grades  of  steel  is  very  nearly  the  same  and  will  be  taken  at 
30,000,000  lbs/in2. 

37.  Elastic  Elongation.  —  As  bearing  upon  deformations  the 
elongation  of  the  steel  at  its  elastic  limit  will  be  here  noted. 
Using  the  above  value  of  the  modulus  of  elasticity  the  elonga- 
tion per  unit  length  of  the  two  grades  of  steel  at  their  elastic 
limit  will  be  as  follows: 

Medium  ......  ......  .......  0.0010-0.0013 

High  ......................  0015-  .0020 


§39.]  ADHESIVE  STRENGTH.  33 

38.  Coefficient  of  Expansion. — The  coefficient  of  expansion 
of  steel  may  be  taken  at  .0000065  per  1°  F. 


PROPERTIES   OF   CONCRETE   AND   STEEL   IN   COMBINATION. 

39.  Adhesion  of  Concrete  and  Reinforcing-bars.  —  The 

high  value  of  the  tangential  adhesion,  or  bond,  of  concrete  to 
steel  rods  embedded  therein  has  long  been  known  and  has 
been  utilized  in  the  placing  of  anchor-rods,  etc.  It  is  some- 
what remarkable,  however,  that  only  recently  has  this  property 
been  made  use  of  in  the  design  of  combination  structural  forms. 
Experience  has  shown  this  adhesion  to  be  sufficiently  reliable 
and  permanent  to  be  utilized  in  such  combination  structures, 
and  plain  smooth  bars  have  been  entirely  successful.  Bars 
of  irregular  section  in  which  adhesion  is  not  entirely  depended 
upon  for  the  bond  are  also  used  to  a  large  extent.  Some  form 
of  mechanical  bond  is  necessary  where  the  adhesion  area  is 
deficient,  and  some  engineers  consider  such  a  bond  desirable  in 
all  cases. 

Numerous  tests  have  been  made  by  various  experimenters 
to  determine  the  adhesion  between  concrete  and  plain  rods 
of  different  forms,  with  results  varying  from  about  200  to 
about  750  lbs/in2.  The  adhesive  strength  is  largely  frictional 
resistance  and  varies  greatly  with  the  roughness  of  the  bars. 
It  also  varies  with  the  quality  of  the  concrete  and  the  method 
of  conducting  the  test.  Usually  the  test  is  made  by  embedding 
the  rod  in  a  block  of  concrete  and  pulling  it  therefrom,  the  rod 
being  stressed  in  tension  and  the  concrete  in  compression. 
This  causes  a  maximum  of  elongation  in  the  steel  at  the  point 
where  it  enters  the  concrete,  while  the  concrete  is  subjected 
to  a  maximum  compression  at  this  same  point.  This  brings 
very  unequal  stresses  upon  the  adhering  surfaces,  tending  to 
a  progressive  separation  until  the  entire  rod  has  started  to  slip, 
after  which  friction  alone  holds  the  rod.  This  unequal  action 
is  greater  the  deeper  the  embedment.  If  the  rod  is  pushed  out, 
both  rod  and  concrete  are  compressed,  although  not  the  same 


34  PROPERTIES    OF    THE   MATERIALS.  [Cn.  II. 

amount  at  the  same  point.  Tests  made  in  this  way  should 
therefore  give  higher  results  than  where  the  rods  are  pulled 
out.  Experimental  results  accord  in  general  with  these  prin- 
ciples. 

Neither  of  these  methods  of  testing  is  entirely  satisfactory. 
What  is  desired  is  the  bond  strength  when  the  concrete  and 
the  rods  are  stressed  as  in  a  beam.  In  this  case  both  the 
concrete  and  the  rod  are  under  tensile  stress  throughout  and 
therefore  subjected  to  deformations  similar  in  character  and 
approximately  equal  in  amount.  They  are  a  minimum  at  the 
end  of  the  beam  and  a  maximum  at  the  centre.  These  con- 
ditions tend  to  make  the  bond  strength  in  a  beam  greater  than 
that  in  the  usual  test  specimen,  but  other  differences  tend  in 
the  opposite  direction.  It  has  been  found,  in  a  series  of  tests 
by  Mr.  Withey  described  further  on,  that  the  local  compression 
to  which  the  concrete  in  the  ordinary  block  specimen  is  subjected 
tends  materially  to  increase  the  bond  strength,  apparently  by 
pressing  the  concrete  more  firmly  against  the  rod.  This  effect 
is  so  marked  in  some  cases  that  the  net  results  from  the  usual 
tests  are  considerably  higher  than  those  made  directly  on 
beams. 

Results  of  Tests  by  Direct  Tension. — Table  No.  4  contains  in 
condensed  form  the  results  of  some  of  the  most  important  tests 
made  by  direct  tension. 

Individual  results  show  little  or  no  effect  due  to  differences 
in  size  of  rod,  but  the  adhesion  for  flat  bars  is  much  less  than 
for  round  or  square  bars. 

In  general  the  stronger  the  concrete  the  greater  the  bond 
strength.  Feret  found  an  increase  of  strength  at  age  of  two 
years  of  about  50%  over  that  at  three  months,  and  a  maximum 
value  for  quite  wet  concrete;  further,  that  a  small  amount  of 
corrosion  increased  the  value.  Bach  found  smaller  values  the 
greater  the  depth,  the  average  value  for  a  4-inch  minimum 
depth  in  1:4  gravel  concrete,  three  months  old,  being  470 
lbs/in2.  He  also  found  greater  values  when  the  rods  were 
pushed  out  than  when  pulled  out 


§39.; 


BOND    STRENGTH. 


35 


TABLE  No.  4. 

BOND  TESTS  BY  DIRECT  TENSION. 


Steel  Rods. 

Authority. 

Kind  of 
Concrete. 

Depth  Em- 
bedded, 
Inches. 

Adhesive 
Strength. 
lbs/in2. 

Size, 

Kind. 

Inches. 

Feret  ; 

1:2:4 

Plain  round 

0.8 

2} 

237 

Ciment  Arme,  p.  755. 

1:2:5 

1  1         1  1 

0.8 

2i 

190 

1:3:4* 

1  1         tt 

0.8 

2> 

237 

1:3:6 

I  (                 t  C 

0.8 

2; 

195 

Emerson  ; 

1:3 

Plain  round 

4 

6 

512 

Eng.  News,  Vol.  LI, 

1:3 

Plain  flat 

Ixi 

6 

293 

1904,  p.  222. 

1:2:4 

Plain  square 

1X1 

10 

587 

1:3:6 

<  <          tt 

1X1 

10 

478 

Talbot; 

1:2:4 

Plain  round 

iand  fc 

6 

438 

Bull.  No.  8,  Univ.  of 

1:2:4 

<  (         it 

\  and  | 

12 

409 

111,  1906. 

1:3:5£ 

ii         1  1 

\  and  f 

6 

364 

1:3:51 

(  C                 (  ( 

£andf 

12 

388 

1:3:5* 

Cold  rolled 

shafting 

1  and  \ 

6 

146 

1:3:5£ 

Mild  steel 

flat 

ftXH 

6 

125 

1:3:6 

Tool-steel 

round 

1 

6 

147 

Withey; 

1:2:4 

Plain  round 

Atol 

6 

400 

Bull.  Univ.  of  Wis., 

1:2:4 

1  1         t  ( 

Atoi 

8 

310 

1907. 

Van  Ornum; 
Eng.  News,  Vol.  LIX, 

1:2:4 
1:2:4 

Plain  round 

«         « 

itoii 
Ho  H 

25  diam. 
40  diam. 

410 
390 

1908,  p.  142. 

Results  of  Tests  on  Beams. — In  an  important  series  of  tests 
by  Mr.  Withey  *  at  the  University  of  Wisconsin,  test  beams 


FIG.  7a. 


were  arranged  as  shown  in  Fig.  la.     The  stresses  in  the  exposed 


Engineering  Record,  1908,  LVII,  p.  798. 


PROPERTIES    OF    THE    MATERIALS. 


[Cn.  II. 


rods  were  determined  by  means  of  extensometers.  The  condi- 
tions were  very  similar  to  those  obtaining  in  an  ordinary  beam, 
but  the  beam  was  prevented  from  failing  by  the  upper  auxiliary 
rods.  Table  No.  4 A  gives  the  principal  results  of  these  tests. 
The  table  also  contains  results  of  comparative  tests  made  at 
the  same  time  by  the  usual  direct  tension  method.  The  last 
column  gives  the  ratio  of  the  two  results.  The  rods  were  of 
ordinary  mild  steel  and  were  free  from  rust. 


TABLE  No.  4A. 

BOND  TESTS  ON  BEAMS. 

UNIVERSITY  OF  WISCONSIN,  1907. 

Concrete  1:2:4;  age  60  days  (Nos.  54-57,  28  days). 


Test  No. 

Diameter  of 
Rod, 
Inches. 

Bond 
Strength. 
lbs/in2. 

Average  of 
Group, 
Ibs/in*. 
(A). 

Average  Bond 
Strength  by 
Direct  Ten- 
sion. lbs/in2. 

Ratio, 
B:  A. 

38 
39 
40 

1     * 

345 
298 
190 

1       278 

394 

1.42 

41 

1 

361 

42 

* 

312 

286 

455 

1.54 

43 

J 

186 

7 

1 

362 

1 

8 
9 

J     * 

264 
201 

|      276 

44 

207 

1 

45 
46 

J 

289 
295 

|      264 

502 

1.90 

47 

136 

] 

48 
49 

1 

174 
180 

163 

487 

2.99 

54 

i 

228 

55 
56 

1 

248 
222 

236 

57 

j 

247 

36 
37 

}     t 

254 

278 

}      266 

467 

1.76 

§39.]  BOND    STRENGTH.  37 

These  tests  indicate  that  the  bond  is  not  affected  by  size 
of  rod  except  in  the  case  of  the  1-inch  size.  This  difference 
is  undoubtedly  due  to  other  factors  not  explained.  Excepting 
the  tests  on  this  size  the  results  are  quite  uniform,  the  average 
of  all  the  60-day  tests  being  275  lbs/in2,  with  maximum  varia- 
tions of  32%  below  and  32%  above  the  average.  The  results 
obtained  by  direct  tension  are  very  much  higher,  averaging 
about  475  lbs/in2,  or  75%  greater.  Other  tests  to  determine 
the  effect  of  age  of  concrete  gave  values  as  follows,  each  result 
being  the  average  from  3  beams;  f-inch  rods  were  used: 

Bond  Strength, 
A8e'  lbs/in* 

7  days 216 

28  days 253 

60  days  . 276 

6  months 316 

It  was  also  found  that  practically  the  same  results  were 
obtained  for  three  different  consistencies,  wet,  medium,  and 
dry,  the  averages  being  respectively  250,  235,  and  275  lbs/in2. 
In  actual  practice  conditions  would  be  less  favorable  to  the  dry 
mixture  owing  to  the  greater  difficulty  of  securing  a  concrete 
free  from  voids. 

Table  4n  contains  results  of  important  tests  on  beams  by 
Bach,*  in  which  the  primary  cause  of  failure  was  the  slipping 
of  the  bars.  Calculating  the  corresponding  bond  stress  by 
eq.  (1)  of  Art.  92,  there  results  the  figures  given  in  the  table. 
The  average  result  for  the  rectangular  beams  with  straight 
rods  only  was  291  lbs/in2,  and  with  stirrups,  330  lbs/in2.  For 
the  T-beams  the  results  for  straight  bars  range  from  158  to 
195  lbs/in2.  The  last  four  beams  contained  bent  bars,  only 
one  bar  being  continued  straight  to  the  end.  Calculating  the 
bond  stress  at  the  end  of  the  beam  with  reference  to  this  single 
rod  gives  an  average  value  of  493  lbs/in.2 

*  Mit.  Ueber  Forsch.  a.  d.  Gebiet  des  Ing.,  1907,  45-47. 


38 


PROPERTIES   OF    THE   MATERIALS, 


[Ce.  II. 


TABLE  No.  4s. 
BOND  TESTS  ON  BEAMS. 

(BACH.) 
Concrete  1:4;  age  6  months;  beams  loaded  at  quarter  points. 


No. 

Kind  of  Beam. 

Reinforcement. 

Calculated 
Bond  Stress  at 
beginning  of 
Slip,  Ibs/in-'. 

Average  for 
Group, 
IDS/  in2. 

2 

312 

1 

3 

Rectangular 

• 
Straight  lods  only 

300 
071 

291 

5 

Beams 

m*t\ 

281 

j 

161 

Straight  rods  and 
stirrups 

}       330* 

330 

162 
163 
164 

Straight  rods  only 
1  Straight  rods  and 
J       stirrups 

158 
1        182 
j       208 

158 
}       195 

165 
166 

T-Beams 

1  straight,  4  bent 
1  straight,  4  bent, 

408 
498 

with  stirrups 

4QO 

167 

1  straight,  4  bent, 

545 

4yo 

with  stirrups 

168 

1  straight,  4  bent 

522 

r  Average  of  three. 

Excepting  where  bent  rods  were  used  these  results  are  about 
the  same  as  those  previously  quoted.  Stirrups  appear  to  in- 
crease the  bond  resistance  somewhat,  an  effect  that  has  been 
reported  by  others.  The  high  results  obtained  on  beams  with 
bent  rods  is  important  in  connection  with  the  design  of  web 
reinforcement,  as  explained  in  Chapter  V.  Inasmuch  as  the 
actual  bond  strength  must  have  been  practically  the  same  as 
in  the  other  tests  these  results  show  that  where  some  of  the 
bars  are  bent  up  so  as  to  reinforce  the  web  of  the  beam  the  bond 
stress  on  the  remaning  straight  bars  near  the  end  of  the  beam 
is  much  less  than  the  theoretical  values  obtanied  by  the  usual 
formulas.  In  this  case]  the  real  bond  stress,  as  shown  by  the 
other  tests,,  appears  to  be  only  about  half  the  calculated  values. 

Frictional  Resistance. — In  bond  tests  it  is  found  that  after 
the  adhesion  has  failed  the  rod  still  offers  much  resistance  to 
movement  due  to  friction  alone.  This  frictional  resistance 


§40.]  BOND    STRENGTH.  39 

varies  from  50  to  about  80%  of  the  initial  bond  strength. 
Hatt  found  a  frictional  resistance,  after  starting,  of  50  to  70% 
of  the  initial  strength,  and  Morsch  reports  such  resistance  as 
about  two-thirds  the  initial.  Talbot  determined  the  frictional 
resistance  in  a  large  number  of  tests,  rinding  it  to  range  quite 
uniformly  from  about  55  to  65%  of  the  initial  bond  strength 
for  both  plain  round  and  flat  rods.  In  the  case  of  cold-rolled 
steel  the  friction  was  only  40%  of  the  bond  strength.  Withey 
found  an  average  result  of  80%  for  6-inch  embedment  and  83% 
for  8-inch  embedment.  Van  Ornum  obtained  average  frictional 
resistances  of  83%  for  mild  steel  and  62%  for  high  steel. 

Conclusions  as  to  Bond  Strength. — A  study  of  the  various 
results  leads  to  the  conclusion  that  for  ordinary  round  or 
square  bars,  not  too  smooth,  the  bond  strength  may  be  taken 
at  from  200  to  300  lbs/in2,  depending  upon  richness  of  mix- 
ture, age  of  cement,  and  roughness  of  bar,  with  a  frictional 
resistance  of  about  two-thirds  this  amount;  a  much  smaller 
value  must  be  taken  for  very  smooth  bars  and  also  for  flat  bars. 

40.  Mechanical  Bond. — The  bond  strength  of  bars  with 
indented  surfaces  depends  upon  the  adhesive  resistance  and 
the  compressive  (or  shearing)  strength  of  the  concrete.  Under 
heavy  stresses  there  is  also  a  tendency  for  the  concrete  to 
split,  owing  to  the  tensile  stresses  developed  by  the  wedging 
action  of  the  bars.  The  initial  movement  in  the  case  of  in- 
dented bars  is  probably  due  to  a  slight  crushing  of  the  concrete 
under  the  projections.  The  bars  cannot  be  pulled  through  the 
concrete,  without  shearing  off  an  area  equal  to  the  total  area 
of  the  indented  portion  and,  in  addition,  overcoming  con- 
siderable friction  or  adhesion.  If  one-half  the  area  is  indented, 
the  ultimate  bond  strength  can  then  be  placed  equal  to  one- 
half  the  shearing  strength  (see  Art.  22),  or  about  one-fourth 
the  compressive  strength  of  the  material.  For  a  1:2:4  con- 
crete this  would  equal  500  to  600  lbs/in2.  In  tests  of  such 
bars,  failures  have  usually  occurred  by  the  splitting  of  the 
specimen  or  the  breaking  of  the  bar,  but  the  results  indicate 
that  the  actual  bond  strength  is  fully  equal  to  these  figures. 


40  PROPERTIES    OF    THE    MATERIALS.  [Cn.  II 

It  appears  from  tests  reported  by  Mr.  T.  L.  Condron  *  that 
the  maximum  bond  strength  is  not  developed  until  a  slight 
movement  has  taken  place.  This  is  particularly  true  with 
deformed  bars.  In  the  case  of  plain  round  and  square  bars 
embedded  from  12  to  38  diameters  the  maximum  strength  was 
very  nearly  reached  under  a  movement  of  1/i0o  inch,  as  measured 
at  the  free  end  of  the  bar.  For  twisted  bars  the  resistance 
continued  to  increase  slightly  under  movements  up  to1/! 6  inch 
or  more,  while  indented  bars  (the  Thacher  and  corrugated  bars) 
showed  steadily  increasing  resistance  under  increased  slip  up  to 
rupture.  The  actual  bond  stress  for  1/ioo  inch  of  movement  wras 
400-600  lbs/in2  for  the  Thacher  and  the  corrugated  bars, 
250-400  lbs/in2  for  the  twisted  bars,  and  175-300  lbs/in2  for 
the  smooth  bars. 

Efficiency  of  Hooked  Ends. — It  is  a  common  practice  among 
designers  to  bend  the  ends  of  reinforcing  rods  into  short  hooks 
generally  consisting  of  right-angle  bends.  The  efficiency  of 
such  hooks  in  increasing  the  bond  strength  has  been  inves- 
tigated by  Bach.  Using  rods  from  J  to  1  in.  in  diameter  and  a 
length  of  bend  of  about  3  diameters  and  an  embedded  length 
of  20  ins.,  he  found  that  the  initial  slip  was  only  slightly 
retarded, — about  as  much  as  would  be  caused  by  the  same 
length  of  straight  bar, — but  the  ultimate  bond  strength  was 
much  increased,  this  increase  averaging  about  50%.  When 
hooked  ends  are  used  they  should  consist  of  bends  through 
180°  with  a  short  length  of  straight  rod  beyond  the  bend.  Such 
hooks  are  found  to  be  very  effective. 

41.  Ratio  of  Moduli  of  Elasticity,  ES:EC. — So  long  as  the 
adhesion  between  steel  and  concrete  is  unimpaired  the  dis- 
tortion of  the  two  materials  will  be  equal.  Their  stresses  will 
then  be  proportional  to  their  moduli  of  elasticity  for  the  load 
in  question,  or  as  the  ratio  of  Es :  EC.  Taking  E8  at  30,000,000 
and  Ec  at  from  2,000,000  to  3,000,000,  the  ratio  will  vary  from 
15  to  10.  In  practice  various  values  of  this  ratio  are  used, 

*  Jour.  West.  Soc.  Eng.,  1907,  Vol.  XII,  p.  100. 


§42.]  EXTENSIBILITY    OF    CONCRETE.  41 

depending  upon  the  kind  of  concrete  and  the  judgment  of  the 
designer.  It  should  also  depend  upon  the  relative  load  for 
which  the  calculations  are  made  and,  to  a  certain  extent,  as 
explained  in  subsequent  chapters,  upon  the  methods  of  calcu- 
lation employed.  As  will  be  seen  in  Chapter  IV,  the  value  15 
corresponds  closely  to  actual  determinations  of  neutral  axes 
in  beams.  It  is  tho  value  commonly  used  in  German  regulations 
and  is  also  specified  in  the  building  laws  of  many  American 
cities. 

Equal  ratios  of  moduli  may  be  assumed  for  both  tension 
and  compression. 

42.  Tensile  Strength  and  Elongation  of  Concrete  when 
Reinforced. — We  have  seen  that  plain  concrete  has  an  ulti- 
mate tensile  strength  of  about  200  lbs/in2  and  a  total  elon- 
gation of  perhaps  Vioooo  part,  corresponding  to  a  value  of 
2,000,000  for  Ec.  Steel  stretches  this  amount  under  a  stress 
of  30,000,000 /1000&=  3000  lbs/in2.  Again,  the  safe  working 
tensile  stress  of  concrete  is  about  50  lbs/in2,  and  if  we  use  a 
value  of  ES/EC=15,  the  corresponding  stress  in  the  steel  will 
be  but  750  lbs/in2.  From  these  relations  it  is  evident  that 
in  reinforced  tension  members  we  must  either  use  very  low  and 
uneconomical  working  stresses  for  steel,  or  else  expect  the 
concrete  to  be  of  no  assistance  in  carrying  stress. 

In  studying  the  behavior  of  reinforced  concrete  under 
tension,  and.  especially  when  constituting  the  tensile  side  of 
a  beam,  results  of  some  experiments  indicate  that  the  con- 
crete in  this  condition  elongates  more  before  final  rupture 
occurs  than  when  not  reinforced,  and  that  the  resistance  of 
the  concrete  is  nearly  constant  and  at  its  maximum  value 
for  some  time  previous  to  rupture.  The  first  to  announce 
this  principle  was  Considere,  whose  tests  indicated  that  the 
ultimate  stretch  of  reinforced  concrete  was  as  much  as  ten 
times  that  of  plain  concrete.  Kleinlogel,*  however,  was  unable 
to  check  these  results,  he  finding  an  elongation  of  practically 

*  Beton  u.  Eisen,  No.2,  1904. 


42  PROPERTIES    OF   THE    MATERIALS.  [On.  II. 

the  same  amount  as  for  plain  concrete.  In  experiments  of 
this  sort  it  is  extremely  difficult  to  determine  just  when  the 
concrete  begins  to  crack.  The  steel  forces  it  to  elongate  prac- 
tically uniformly,  even  after  rupture  begins,  so  that  a  crack 
will  open  up  very  slowly  and  will  therefore  remain  almost 
invisible  for  some  time. 

In  some  experiments  made  at  the  University  of  Wisconsin 
in  1901-2  a  very  delicate  method  of  detecting  incipient  cracks 
was  accidentally  discovered.  It  was  found  that  beams  cured 
in  water  which  were  only  partially  dried  before  testing 
would,  when  tested,  show  very  fine  hair-cracks  at  an  early 
stage,  and  moreover,  by  watching  closely,  it  was  observed  that 
preceding  the  appearance  of  a  crack  there  would  appear  a 
dark  wet  line  across  the  beam.  Such  a  line  would  soon  be 
followed  by  a  very  fine  crack.  A  larger  series  of  tests  were  under- 
taken in  the  following  year  by  a  different  set  of  experimenters, 
who  observed  the  same  phenomenon.  Careful  measurements  of 
extension  showed  that  these  streaks  or  "water-marks ",  as 
they  were  named,  occurred  at  practically  the  same  deformation 
at  which  the  concrete  ruptured  .when  not  reinforced.  Some 
of  the  results  are  given  in  Table  No.  5.*  The  beams  were  of 
1:2:4  mixture  by  weight  and  were  6"X6"  in  cross-section  by 
60  inches  span. 

That  these  water-marks  were  incipient  cracks  was  deter- 
mined by  sawing  out  a  strip  of  concrete  along  the  outer  part 
of  the  beam.  Fig.  8.  is  a  photograph  showing  the  results  of 
this  experiment.  Very  close  observation  also  in  many  cases 
showed  hair-like  cracks  appearing  very  soon  after  the  appear- 
ance of  the  water-marks. 

Comparing  the  observed  and  calculated  elongations  of  the 
reinforced  concrete  with  those  of  the  plain  concrete  at  rupture 
it  will  be  seen  that  the  initial  cracking  in  the  former  occurs  at 
an  elongation  practically  the  same  as  reached  by  the  latter  at 
rupture. 

*  Bulletin  No.  4,  Engineering  Series,  Univ.  of  Wis.,  1906. 


§42.] 


EXTENSIBILITY  OF  CONCRETE. 


43 


TABLE  No.  5. 

TESTS  OF  BEAMS  SHOWING  EXTENSIBILITY  OF  CONCRETE. 


Proportionate  Extension. 

Compressive 

No. 

Age. 

Method 
of  Loading. 

Strength  of 
Cubes, 
lbs/in2. 

At  First 
Water- 

At First 
Visible 

mark. 

Crack. 

8 

3  months 

At  third  points 

.00011 

.00064 

4250 

10 

n 

<( 

.00024 

.00046 

2500 

22 

(  t 

^  e 

.00025 

.00065 

2775 

26 

1  1 

•• 

.00016 

.00056 

3000 

30 

1  1 

1  1 

.00012 

.00064 

2600 

7 

1  month 

At  center 

.00015 

.00036 

3500 

5 

t  { 

a 

.00020 

.00031 

3500 

13 

tt 

il 

.00009 

.00011 

2350 

23 

a 

1  1 

.00020 

.00060 

2500 

35 

ii 

i  C 

.00013 

.00053 

3150 

2* 

1  1 

(e 

At  rupture 

.00013 

3000 

1* 

(  ( 

ii 

.00010 

2500 

*  Nos.  2  and  1  were  plain  concrete  beams.  The  extensions  of  the  beams  loaded  at 
the  third  points  were  measured  by  extensometers;  those  of  the  center  loaded  beams  were 
calculated  from  deflections. 


FIG.  8. 


44  PROPERTIES    OF    THE    MATERIALS.  [Cn.  II. 

Since  these  experiments  were  made  a  very  careful  series 
of  observations  have  been  made  by  Bach.*  He  noted  the 
sama  "  water-marks/'  and  his  series  of  tests  on  85  beams  of 
rectangular  and  T-section,  of  1:4  concrete  from  6  to  8  months 
old,  gave  very  uniform  results.  Water-marks  appeared  at 
elongations  of  from  .00006  to  .00010  part.  He  was  also  able 
to  detect  these  on  plain  concrete  beams  at  the  same  elonga- 
tion and  at  loads  about  80%  of  the  breaking  load;  and  on  several 
tension  specimens  he  found  the  elongation,  just  previous  to 
rupture,  to  b3  .00006  to  .00010  part.  He  was  able  to  observe 
the  first  well-defined  crack  at  elongations  of  .00012  to  .00014. 
He  concludes  that  reinforced  concrete  will  begin  to  crack  at 
the  sam3  elongation  as  plain  concrete. 

It  should  ba  said  that  in  many  cases  the  first  " water-marks" 
do  not  extend  entirely  across  the  tension  face  of  the  beam, 
so  that  the  concrete  as  a  whole  possesses  some  tensile  strength. 

The  presence  of  these  minute  cracks  of  course  seriously 
affects  tli3  tensile  strength  of  the  concrete,  and  as  they  appear 
at  an  elongation  corresponding  to  a  stress  in  the  steel  of  5000 
Ibs/in2  or  less,  it  would  seem  that  no  allowance  should  be 
made  for  the  tensile  resistance  of  the  concrete  where  the  usual 
working  stresses  are  used  for  steel.  In  some  cases,  however, 
the  stresses  in  the  steel  are  necessarily  very  low,  in  which  case 
it  may  be  proper  to  consider  the  tensile  resistance  of  the  con- 
crete. This  limit  may  be  placed  at  about  2000  Ibs/in2,  corre- 
sponding to  an  elongation  of  .00006  part  and  a  stress  of  150 
to  175  Ibs/in2  in  the  concrete. 

In  practical  design  the  most  important  question  which 
arises  is  how  far  a  concrete  may  be  cracked  without  exposing 
the  steel  to  corrosive  influences.  In  this  respect  experience 
indicates  that  the  minute  cracks  which  appear  in  the  early 
stages  of  the  tests  are  of  no  practical  consequences. 
4  43.  Relative  Contraction  and  Expansion. — Temperature 
changes  affect  both  the  steel  and  the  concrete.  But  as  the 
coefficient  of  expansion  of  steel  is  .0000065  and  of  concrete 

*  Zeit.  Ver.  Dt.  Ing.,  1907. 


§43.]  CONTRACTION  AND  EXPANSION.  45 

.000006,  the  two  materials  will  be  but  slightly  stressed  because 
of  any  difference  in  their  rates  of  expansion. 

The  effect  of  shrinkage  in  hardening  is  more  serious.  As 
shown  in  Art.  29,  the  hardening  of  concrete  is  accompanied 
by  more  or  less  contraction  if  in  air,  or  expansion  (to  a  less 
degree)  if  in  water.  Concrete  which  is  unrestrained  either  by 
steel  reinforcement  or  by  exterior  attachment  will  shrink  or 
swell  proportionally  and  no  stresses  will  thereby  be  developed. 
If  restrained  by  reinforcing  material  only,  a  shrinkage  will 
develop  tensile  stresses  in  the  concrete  and  compressive  stresses 
in  the  steel. 

If  it  be  assumed  that  concrete  when  reinforced  tends  to 
shrink  the  same  amount  as  plain  concrete,  and  that  such  shrink- 
age is  prevented  only  so  far  as  the  stresses  developed  in  the 
steel  react  upon  the  concrete  and  cause  an  opposite  movement, 
then  it  will  be  found,  using  the  ordinary  values  of  the  modulus 
of  elasticity,  that  the  stresses  developed  in  both  the  concrete 
and  the  steel  will  be  large.  These  stresses  would  be  determined 
as  follows: 

Let  c=  coefficient  of  contraction  of  the  concrete; 

/c=unit  stress  in  concrete  (tensile); 

fa  =  unit  stress  in  steel  (compressive)  ; 

p=  steel  ratio; 

n  =  Es/Ec. 

Then  the  net  contraction  per  unit  length  as  measured  by 
the  concrete  will  be  c—fc/Ec,  and  as  measured  by  the  steel  will 
be  }8/E8.  These  values  are  equal.  Also,  for  equilibrium,  fc  =  pfa. 
From  these  equations  we  get 


and 

/•-£    •  •  •  ......  <* 

If,  for  example,  c=.0003,  Ec=  2,000,000,  n=15,  p=l%,  then 
/c=80  lbs/in2  tension  and  /s  =  8000  lbs/in2  compression.  If 
p=2%,  /C=I40  and  A  =7000  Ibs/m2. 


46  PROPERTIES    OF    THE    MATERIALS.  [Cn.  II. 

It  is  doubtful  if  such  large  initial  stresses  actually  occur  in 
reinforced  concrete  due  to  shrinkage  in  hardening. 

The  experiments  of  Considere  on  the  actual  contraction 
of  reinforced  concrete,  already  quoted  in  Art.  29,  indicate 
that  the  deformation  is  less  than  the  above  theory  would  call 
for.  For  example,  the  observed  contraction  of  .01%  in  rein- 
forced mortar  would  call  for  a  stress  in  the  steel  of  only  about, 
3000  lbs/in2,  and  in  the  concrete  of  only  30  to  60  lbs/in2.  In 
slowly  hardening,  with  the  steel  in  place,  there  is  probably  a 
gradual  adjustment  in  the  concrete  which  results  in  less  internal 
stress  than  the  experiments  on  plain  concrete  would  indicate. 
Where  the  structure  is  restrained  by  outside  supports  which  are 
relatively  more  rigid  than  the  reinforcing  steel,  the  stresses  in 
the  concrete  become  greater  and  may  easily  reach  the  limit 
of  the  tensile  strength,  thus  causing  cracks.  (For  further 
discussion  of  reinforcement  under  such  conditions,  see  Chapter 
V,  Art.  142.) 


CHAPTER  III. 

GENERAL  THEORY. 

44.  Kinds  of  Members.— Structural  .members  are,  for  con- 
venience,  usually  divided  into  tension  members,   compression 
members,  and  beams,  according  as  the  forces  to  be  resisted 
produce  in  the  member  simple  tension,  simple  compression, 
or  simple  bending.    Bending  moment  is  often  accompanied 
by  tension  or  compression,  producing  what  are  called  combined 
stresses  of  bending  and  tension,  or  bending  and  compression. 
Since  reinforced  concrete  is  not  used  for  plain  tension  mem- 
bers the  analysis  will  be  confined  to  the  beam,  both  under 
plain  bending  and  under  combined  stresses,  and  to  the  com- 
pression member  or  column.    The  flat  slab  supported  on  four 
sides  will  be  considered  as  a  special  case  of  beam.    In  rein- 
forced-concrete  construction  the  beam  is  the  most  important 
element,  being  used  under  a  great  variety  of  conditions. 

45.  Relation  of  Stress  Intensities  in  Concrete  and  Steel. 
In  the  following  discussion  it  will  be  assumed  that  the  con- 
crete and  steel  adhere  perfectly  and  therefore  deform  equally. 
Nearly  all  reinforced-concrete  construction  is  dependent  upon 
this  equal  action  of  the  two  materials,  although  simple  adhesion 
is  not  always  entirely  depended  upon.    Many  types  of  deformed, 
or  roughened,  bars  are  used  so  as  to  give  the  steel  a  grip  inde- 
pendent of  the  adhesion,  and  in  other  cases  bars  are  bent  or 
anchored  at  the  ends,  but  in  all  cases  it  is  assumed  that  the 
materials    adhere    perfectly    and    therefore    deform    equally. 
Many  tests  show  that  under  proper  design  this  is  for  all  prac- 
tical purposes  true. 

47 


48  GENERAL  THEORY.  [Cn.  III. 

Since  the  modulus  of  elasticity  of  a  material  is  the  ratio 
of  stress  to  deformation,  it  follows  that  for  equal  deformations 
the  stresses  in  different  materials  will  be  as  their  moduli  of 
elasticity.  If 

/s=unit  stress  in  steel, 
/c=unit  stress  in  concrete, 
Es= modulus  of  elasticity  of  steel,  and 
Ec  =  modulus  of  elasticity  of  concrete, 
we  have  the  fixed  relation 

fJfc=Es/Ec .    (1) 

46.  Distribution  of  Stress  in  a  Homogeneous  Beam. — To 

assist  in  forming  correct  notions  of  the  action  of  steel  reinforce- 
ment in  a  concrete  beam,  it  will  be  desirable  to  consider,  at 
the  outset,  the  nature  of  the  stresses  due  to  bending  moment 
in  a  plain  concrete  or  homogeneous  beam  of  any  material. 
Considering  a  vertical  section  at  any  point  there  will  exist 
in  general  certain  normal  stresses  (tensile  and  compressive) 
and  certain  tangential  or  shearing  stresses.  A  knowledge  of 
these  stresses  on  a  vertical  section,  together  with  the  well-known 
principle  that  the  shearing  stress  at  any  point  is  of  equal  in- 
tensity vertically  and  horizontally,  is  sufficient  for  the  design- 
ing of  ordinary  beams. 

In  accordance  with  the  common  theory  of  flexure,  the 
normal  stress  on  a  vertical  section  varies  in  intensity  as  the 
distance  from  the  neutral  axis,  and  therefore  the  variation  is 
represented  by  the  ordinates  to  a  straight  line  as  in  Fig.  9. 

The  shearing-stress  intensity  is  a  maximum  at  the  neutral 
axis  and  is  zero  at  the  outer  fibres.  At  any  given  point  in  the 
section  it  is  given  by  the  equation 

v=VS/Ib, (1) 

in  which  V  denotes  the  entire  shear  at  the  section  containing 
the  point  under  consideration,  /  the  moment  of  inertia  of  the 
section  with  respect  to  the  neutral  axis,  6  the  breadth  of  the 
section  at  the  point,  and  S  the  statical  moment  of  the  part  of 


46.] 


GENERAL  STRESS  DISTRIBUTION. 


49 


the  section  above  (or  below)  the  point  with  respect  to  the 
neutral  axis.  For  a  rectangular  beam  the  intensity  of  shear 
varies  as  the  ordinates  to  a  parabola,  as  shown  in  Fig.  10,  the 
maximum  value  being  3/2  times  the  average,  or  equal  to 
3  V_ 
2'bd' 

If  the  stresses  on  inclined  planes  are  analyzed,  it  is  found 
that  the  normal  and  shearing  stresses  will  not  be  the  same  as 
on  vertical  planes;  and,  furthermore,  that  wherever  shearing 
stress  exists  on  a  vertical  plane  the  maximum  normal  stress 


FIG.  9.  FIG.  10. 

will  not  be  on  a  vertical  section,  but  on  an  inclined  one.  It  is 
proved  in  treatises  on  mechanics  that  if  /-represents  the  hori- 
zontal unit  tensile  stress  and  v  the  vertical  or  horizontal  unit 
shearing  stress  at  any  point  in  a  beam,  the  maximum  tensile 
stress  will  be  given  by  the  formula 

/.=J/+\/i/2  +  ^, (2) 

I 

and  the  direction  of  this  maximum  tension  is  given  by  the 
formula  tan  2d  =  2v/f,  where  6  is  the  angle  of  the  maximum 
tension  with  the  horizontal. 

A  study  of  these  formulas  shows  that  at  all  points  in  a  beam 
where  the  shear  is  zero,  the  direction  of  the  maximum  tension 
is  horizontal,  as  at  points  of  maximum  bending  moment  and 
along  the  outer  fibres  of  the  beamJ^Vherever  the  horizontal 
fibre  stress  is  zero  (at  the  neutral  surface  and  at  all  sections  of 
zero  bending  moment),  the  direction  of  the  maximum  tension 
is  inclined  45°  to  the  horizontal,  and  its  intensity  is  equal  to 
the  unit  shearing  stress  at  the  same  place.  Above  the  neutral 
axis  of  a  section  where  the  bending  moment  is  not  zero,  the 
inclination  of  the  maximum  tension  is  greater  than  45°,  becom- 


50 


GENERAL  THEORY. 


[Cn.  Ill 


ing  90°  at  the  upper  or  compressive  fibre.  Fig.  11  illustrates 
the  variation  in  normal  stress,  shearing  stress,  and  maximum 
tensile  stress  throughout  the  entire  depth  of  a  rectangular  beam. 
The  outer  normal  or  fibre  stress  is  assumed  at  200  lbs/in2,  and 
the  shearing  stress  at  the  neutral  axis  at  150  lbs/in2.  The 


Compression 


FIG. 


B  Tension  E 

o  —  Showing  Variation  of  Intensities  of  Normal  Stress,  Shear,  and 
Maximum  Tension. 


variation  in  the  fibre  stress  is  shown  by  the  straight  line  DE, 
and  that  in  the  shearing  stress  by  the  parabolic  curve  A  CB. 
By  means  of  eq.  (2)  the  maximum  tensile  stresses  have  been 
computed  ;  these  are  represented  by  the  line  AHCJE. 

Fig.  12  illustrates  the  direction  of  the  maximum  tensile 


FIG.  12. — Lines  of  Maximum  Tension. 

stresses  in  a  rectangular  beam.  The  exact  direction  at  any 
point  depends  upon  the  relation  between  shear  and  bending 
moment.  Lines  of  maximum  compression  would  run  at  right 
angles  to  the  lines  shown  and  lines  of  maximum  shear  at  angles 
of  45°  therewith. 

47.  Purpose  and  Arrangement  of  Steel  Reinforcement— 
The  purpose  of  steel  reinforcement  is  to  carry  the  principal 
tensile  stresses,  the  concrete  being  depended  upon  for  the  com- 


I  48.]          COMMON  THEORY  OF  BEAMS.  51 

pressive  and  shearing  stresses,  its  resistance  to  such  stresses 
being  large.  If  no  steel  were  present  the  concrete  would  tend 
to  rupture  on  lines  perpendicular  to  the  direction  of  maximum 
tension,  as  shown  in  Fig.  12,  and  hence  we  may  conclude  that 
the  ideal  tension  reinforcement  would  require  the  steel  to  be 
distributed  in  the  beam  along  the  lines  of  maximum  tension. 
At  the  centre  of  the  beam,  or  place  of  maximum  moment,  this 
directlonKis  horizontal  for  the  entire  depth  of  the  beam,  and 
horizontal  rods  placed  near  the  lower  edge  of  the  beam  con- 
stitute proper  and  sufficient  reinforcement.j  As  we  approach 
the  ends  of  the  beam,  where  the  shear  is  large,  the  intensity  of 
the  inclined  tensile  stresses  becomes  of  importance,  and  in  many 
cases  these  stresses  require  special  attention.  Horizontal  rods 
at  the  bottom  are  still  necessary,  but  do  not  entirely  reinforce 
the  concrete  against  'tension,  so  that  special  consideration  must 
be  given  to  reinforcement  in  the  body  of  the  beam.  The 
arrangement  of  this  reinforcement  demands  careful  consid- 
eration. 

For  purposes  of  discussion,  the  subject  of  beams  will  first  be 
treated  with  reference  only  to  the  horizontal  reinforcement. 
The  inclined  tensile  stresses  will  be  considered  separately. 

48.  The  Common  Theory  of  Flexure  and  its  Modifi- 
cation for  Concrete.— ^-The  common  theory  of  flexure  is  based  on 
two  main  assumptions,  namely,  (1)  a  plane  cross-section  of  an  un- 
loaded beam  wrill  still  be  plane  after  bending  (Navier's  hypoth- 
esis) ;  (2)  the  material  of  the  beam  obeys  Hooke's  law,  which 
is,  briefly  stated,  " stress  is  proportional  to  strain7'.  From  the 
first  assumption  it  follows  that — The  unit  deformations  of 
the  fibres  at  any  section  of  a  beam  are  proportional  to  their  dis- 
tances from  the  neutral  surface."  In  the  case  of  simple  bending 
(all  forces  at  right  angles  to  the  beam)  the  neutral  axis  lies  at 
the  centre  of  gravity  of  the  section;  in  the  case  of  bending 
combined  with  direct  tension  or  compression,  the  neutral  axis 
may  lie  in  the  section  or  be  merely  an  imaginary  line  without 
the  section.  'From  the  second  assumption  it  follows  that— 
The  unit  stresses  in  the  fibres  at  any  section  of  a  beam  also  are 


52  GENERAL  THEORY.  [CH.  III. 

proportional  to  the  distances  of  the  fibres  from  the  neutral  surface. 
This  may  be  called  the  linear  law  of  the  distribution  of  stress.  - 

The  linear  law  is  the  basis  of  all  practical  flexure  formulas 
excepting  some  for  reinforced-concrete  beams.  It  is  true  that 
wrought  iron  and  steel  are  the  only  important  structural  mate- 
rials which  closely  obey  Hooke7s  law,  and  they  only  within 
their  elastic  limits.  But  under  working  conditions  these  mate- 
rials are  not  stressed  beyond  these  Limits,  and  so  the  formulas 
ordinarily  hold.  Timber,  stone,  and  cast  iron  can  hardly  be 
said  to  obey  Hooke's  law,  yet  for  working  conditions  the  com- 
mon flexure  formulas  for  these  materials  are  roughly  correct 
and  they  are  in  general  use. 

In  the  case  of  those  materials  which  do  not  obey  Hooke's 
law,  as  concrete,  and  for  all  materials  when  stressed  beyond 
their  elastic  limit,  the  common  theory  does  not  strictly  apply. 
An  exact  analysis  requires  the  use  of  the  actual  tension  and 
compression  stress-strain  diagrams  for  the  materials  up  to  the 
limit  of  the  actual  stresses  involved.  It  will  be  assumed  still 
that  plane  sections  remain  plane  during  bending  so  that  defor- 
mations will  be  proportional  to  the  distances  of  the  fibres  from 
the  neutral  surface.  The  experiments  by  Talbot,*  though  not 
conclusive,  bear  out  this  assumption  in  the  more  important 
case  of  reinforced  beams.  Experiments  by  Schiile,f  however, 
seem  to  show  that  original  plane  sections  do  not  remain  plane. 
Nevertheless  Navier's  hypothesis  will  probably  remain  a  basis 
of  flexure  formulas  for  reinforced-concrete  beams. 

The  variation  of  the  normal  stress  on  the  cross-section  can 
then  be  represented  graphically  in  the  following  manner:  Let 
Fig.  13a  be  the  stress-strain  diagram,  compression  above  the 
x  axis  and  tension  below,  for  the  material  in  question  as 
determined  by  direct  compression  and  tension  tests.  These 
curves  are  plotted  with  unit  stresses  as  abscissas  and  unit 
strains  as  ordinates.  Let  Fig.  13&  represent  the  beam,  cut 

*  Univ.  of  111.  Bull.,  Vol.  II,  No.  1,  p.  28. 

f  Mitteilungen  der  Materialpriifungs-Anstalt  am  Polytechnikum  in  Zurich, 
Vol.  X  (1906),  p.  40. 


§49.] 


COMMON  THEORY  OF  BEAMS. 


53 


on  section  AB  where  the  stresses  are  to  be  investigated. 
The  neutral  axis  is  at  N.  Since  the  deformations  of  the  fibres 
are  proportional  to  the  distances  of  the  fibres  from  the  neutral 
axis,  these  distances  themselves,  Nl,  N2,  N3,  etc.,  will  represent 
to  some  scale  the  deformations.  If  the  unit  deformation  at 
point  1  is  then  represented  by  Nl  the  corresponding  stress  can 
be  determined  from  the  diagram  of  Fig.  13a,  using  the  proper 
scale  in  both  cases.  Lay  off  the  distance  la  to  represent 
that  stress.  Proceeding  similarly  for  all  points  and  connecting, 
we  have  the  stress  curve  A'NB',  which  is  nothing  more  than 


v  B 


FIG.  13. 


a  portion  of  the  diagram  of  Fig.  13a  plotted  to  a  different 
scale. 

49.  Resisting  Moment  and  Inefficiency  of  Concrete 
Beams.  —  For  use  in  the  following  and  other  discussions  on 
flexure  three  important  principles  from  the  mechanics  of  beams 
are  now  recalled: 

(1)  For  beams  rectangular  in  section,   the  average   unit 
tensile  and  compressive  fibre  stresses  on  any  cross-section  are 
represented  by  the  average  abscissas  in  the  tensile  and  com- 
pressive parts  of  the  stress  diagram,  NBBf  and  NAA',  respect- 
ively (Fig.  136).      Also  the  whole  tension  T  and  whole  com- 
pression C  on  the  cross-section  are  proportional  to  the  areas 
NBB'  and  NAA']    hence,  according  to  some  scale,  the  areas 
represent  T  and  C  respectively. 

(2)  The  resultant  tension  T  and  resultant  compression  C 
act  through  the  centroids  of  the  tensile  and  compressive  areas 
in  the  stress  diagram. 

(3)  When  all  the  forces  (loads  and  reactions)  applied  to 


54 


GENERAL  THEORY. 


[Cn.  III. 


the  beam  act  at  right  angles  to  it,  then  the  resultant  tension  T 
equals  the  resultant  compression  C;  hence  the  two  stresses 
constitute  a  couple  —  "the  resisting  couple". 

Fig.  14  is  a  stress-strain  diagram  of  a  gravel  concrete  for 
both  tension  and  compression.  For  any  section  of  a  beam 

made  of  this  concrete,  the  stress 

/ 

diagram  is  a  certain  part  of 
the  stress-strain  diagram,  the 
exact  part  depending  on  the 
loading.  Suppose  that  the  loads 
produce  in  the  lower  fibre  at 
the  section  in  question  a  unit 
stress  represented  by  W  say, 
then  T  is  represented  by  NW 
and  C  by  an  area  Naa'  deter- 
mined from  the  principle  that 
FlG  14  it  must  equal  the  area  Nbb'. 

Hence   the  stress   diagram   is 

aa'Nb'b,  and  the  unit  stress  on  the  upper  fibre  is  represented 
by  aa'.  Furthermore,  ab  represents  the  depth  of  the  beam, 
and  N  the  position  of  the  neutral  axis.  Likewise,  when  the 
unit  stress  on  the  lower  fibre  is  BB'  (the  ultimate  tensile  strength) 
and  the  beam  is  on  the  point  of  failing,  T  is  represented  by 
the  area  NBB',  and  C  by  the  equal  area  NAA'-,  hence  the  stress 
diagram  for  the  failure  stage  is  AA'NB'B,  and  the  unit  stress 
on  the  upper  fibre  is  A  A'. 

50.  Resisting  Moment.—  The  resisting  moment  of  a  section 
is  the  moment  of  the  resisting  couple  which  acts  at  that 
section.  Its  value  is  the  product  of  the  tension  (or  com- 
pression) and  the  distance  between  the  centroids  of  these 
stresses.  For  example,  at  the  failure  stage  of  the  beam 
above  referred  to  the  average  unit  tensile  stress  scales 
128  lbs/in2,  and  F£  =  0.615=  0.6d,  d  denoting  depth  of 
beam.  Hence  if  b  denotes  the  breadth  of  the  section, 


§  51.]          COMMON  THEORY  OF  BEAMS.  55 

The  vertical  distance  between  the  centroids  of  the  shaded 
parts  (NAA'  and  NBBf)  of  the  diagram  is  QMAB-,  hence 
the  arm  of  the  resisting  couple  is  0.64d,  and  the  computed 
ultimate  resisting  moment  of  a  beam  made  of  the  concrete 
under  consideration  is  76.8frdx0.64<i=49.2&d2  in-lbs.;  6  and  d 
to  be  expressed  in  inches. 

Partly  to  test  the  correctness  of  the  theory  of  flexure  of 
concrete  beams,  Professor  Morsch*  made  three  beams  15X20  cm. 
in  section  and  several  tension  and  compression  specimens  of 
the  same  mix  of  concrete.  From  tests  on  the  specimeng  he 
obtained  a  stress-strain  diagram  from  which  he  computed 
the  probable  resisting  moment  of  the  beams  to  be  3.456d2= 
3.45  X 15  X202  =  20,700  kg-cm.  The  average  of  the  actual 
resisting  moments  of  the  beams  (determined  from  tests  to 
destruction)  was  22,100  kg-cm.,  an  agreement  to  be  regarded 
as  highly  satisfactory. 

The  working  resisting  moment  of  a  rectangular  beam  can 
be  computed  from  the  stress-strain  diagram  for  the  material 
in  this  same  manner.  Fortunately,  engineers  are  not  called 
upon  to  compute  resisting  moments  by  this  method.  It  is 
here  set  forth  principally  as  a  means  of  introducing  important 
ideas  bearing  on  reinforced-concrete  beams. 

51.  Inefficiency  of  Concrete  Beams. -AWhen  a  beam  of  the 
concrete  above  referred  to  is  loaded  to  the  breaking  point, 
the  greatest  unit  compressive  stress  in  the  beam  is  the  stress 
AA'j  which  is  in  this  case  about  375  lbs/in2.  This  is  very  low 
compared  to  the  ultimate  compressive  strength  (2500  lbs/in 2)> 
and  the  difference  indicates  a  wasteful  use  of  concrete. 

The  unshaded  portion  of  the  stress-strain  diagram  (Fig.  14) 
is  also  significant  in  this  connection,  for  it  indicates  the  unused 
compressive  strength  of  the  concrete  above  the  neutral  surface 
when  the  tensile  strength  of  that  below  is  fully  developed 
and  the  beam  is  about  to  fail. 

Another  way  to  express  the  inefficiency  of  a  concrete  beam 

*  Der  Eisenbetonbau. 


56  GENERAL  THEORY.  [Cn.  III. 

is  to  compare  its  ultimate  resisting  moment  with  that  which 
it  would  have  if  the  tensile  strength  and  elastic  properties 
were  the  same  as  the  compressive.  On  this  supposition  the 
tensile  stress-strain  diagram  would  be  like  the  compressive; 
and  for  the  concrete  of  Fig.  14,  the  ultimate  C  and  T  are  rep- 
resented by  the  area  NYY',  and  the  arm  of  the  resisting  couple 
by  twice  the  vertical  distance  of  the  centroid  of  the  area  NYY' 
above  N.  Actual  measurement  of  the  area  and  distance  gives 
0=775bd  and  arm=0.64d;  hence  the  ideal  ultimate  resisting 
moment  is  775&dx0.64d=496fcd2  as  against  49.26d2,  the  actual 
value. 

{  To  supply  the  deficiency  in  tensile  strength  of  concrete  is 
the  main  purpose  of  steel  reinforcement.  A  comparatively 
small  amount  of  steel  (rods  or  bars  whose  combined  sectional 
area  is  from  1  to  2  per  cent  of  the  total  sectional  area  of  the 
beam)  properly  embedded  will  so  strengthen  the  tensile  side 
of  the  beam  that  the  great  strength  of  the  compressive  side 
can  be  utilized.  The  exact  amount  of  steel  required  in  any  case 
depends  on  the  elastic  properties  of  the  concrete  and  steel. 
52.  Varieties  of  Flexure  Formulas. — Many  formulas  have 
been  proposed  for  the  strength  of  reinforced-concrete  beams. 
The  differences  among  them  arise  principally  from  three  sources, 
namely:  (1)  The  method  of  applying  the  factor  of  safety, 
(2)  the  law  of  distribution  of  the  compressive  fibre 'stress  in 
the  concrete,  and  (3)  the  value  of  the  tensile  fibre  stress  in 
the  concrete.  In  regard  to: 

(1)  Two  views  are  held  as  to  the  proper  method  cf  applying 
the  factor  of  safety.  For  example,  to  ascertain  the  safe  load 
for  a  given  beam,  some  engineers  assume  working  strengths  for 
the  concrete  and  steel,  with  which,  by  means,  of  a  suitable 
flexure  formula,  they  compute  the  safe  load  directly;  other 
engineers  compute  the  breaking  load  of  the  beam  by  a  suitable 
formula  and  then,  with  reference  to  this  load,  they  decide 
upon  the  safe  load.  (The  pros  and  cons  of  these  two  methods 
are  discussed  in  Art.  118.)  Formulas  for  working  conditions 
(for  use  in  the  first  method)  are  explained  hi  Arts.  54-9;  those 


§52.] 


COMMON  THEORY  OF  BEAMS. 


57 


for  ultimate  conditions   (for  use  in  the  second  method)   in 
Arts.  60-4;   and  those  for  both  conditions  in  Arts.  65-70. 

(2)  As  already  explained  in  Art.  48,  the  distribution  of  the 
compressive  fibre  stress  can  be  represented  by  a  portion  of 
the  stress-strain  diagram  for  the  concrete.  As  shown  in 
Art.  23,  the  stress-strain  curve  for  concrete  up  to  and  even 
beyond  working  stresses  is  nearly  straight,  and  the  most  widely 
used  flexure  formulas  for  working  conditions  are  based  on  the 
assumption  that  the  stress-strain  curve  is  practically  straight 
up  to  working  stresses.  Formulas  of  Arts.  54-9  and  all  other 
flexure  formulas  of  this  book  (except  those  of  Arts.  60-70) 
are  based  on  this  assumption.  When  the  curvature  of  the 
stress-strain  curve  has  been  taken  into  account,  it  has  gen- 
erally been  assumed  to  be  an  arc  of  a  parabola,  the  vertex 


FIG.  15. — Distribution  of  Fibre  Stress  in  Concrete  According  to  Various 

Assumptions. 

being  taken,  by  some,  at  the  end  of  curve  (the  ultimate  strength 
end)  and,  by  others,  beyond  that  point.  The  formulas  of  Arts. 
60-70  are  based  on  a  parabolic  stress-strain  curve,  the  vertex 
being  at  the  end. 

(3)  As  explained  in  Art.  42,  when  a  reinforced-concrete 
beam  is  being  loaded,  the  concrete  adjoining  the  steel  fails 
(cracks)  probably  always  before  the  stress  in  the  steel  reaches 
5000  lbs/in2,  and  when  the  stress  reaches  working  values  the 
cracks  will  have  extended  well-nigh  to  the  neutral  surface. 
The  amount  of  tension  remaining  in  the  concrete  at  the  section 


58  GENERAL  THEORY.  H.  III. 

of  the  crack  is  comparatively  small,  and  this  tension  being 
near  the  neutral  surface,  the  resisting  moment  due  to  it  is 
also  small  compared  to  that  due  to  the  tension  in  the  steel. 
In  a  certain  formula  for  ultimate  resisting  moment  in  which 
this  residual  tension  in  the  concrete  is  allowed  for,  the  value 
of  the  term  expressing  the  contribution  of  this  tension  is  less 
than  J  per  cent  of  the  total  moment.  It  is  the  almost  universal 
practice  to  neglect  tKis  tension  entirely  in  flexure  formulas; 
this  practice  is  followed  in  this  book. 

An  idea  of  the  variety  of  flexure  formulas  proposed  can  be 
gained  from  Fig.  15,  which  shows  nine  distributions  of  fibre 
stress  in  the  concrete  according  to  as  many  different  formulas. 
53.  Notation. — Fuller  explanations  of  some  of  these  sym- 
bols are  given  in  subsequent  articles  where  the  formulas  are 
derived;  see  also  Fig.  16. 

fa  denotes  unit  fibre  stress  in  steel; 

fc      "  ts     ft  concrete  at  its  compressive  face; 

ea       "        "    elongation  of  the  steel  due  to  fs; 
ec       "         "    shortening  of  the  concrete  due  to  /c; 
E8       "      modulus  of  elasticity  of  the  steel; 
Ec       "  "      "  the  concrete  in  compression; 

n       "      ratio  Ea/Ee; 

T       "      total  tension  in  steel  at  a  section  of  the  beam; 
C       "      total  compression  in  concrete  at  a  section  of  the 

beam; 

M8       "      resisting  moment  as  determined  by  steel; 
Mc       "      resisting  moment  as  determined  by  concrete; 
M       "      bending  moment  or  resisting  moment  in  general; 
b       t(       breadth  of  a  rectangular  beam; 
d       "      distance  from  the  compressive  face  to  the  plane 

of  the  steel; 
k       "      ratio  of  the  depth  of  the  neutral  axis  of  a  section 

below  the  top  to  d\  . 

/       "      ratio  of  the  arm  of  the  resisting  couple  to  d; 
A       "      area  of  cross-section  of  steel; 
p       "      steel  ratio,  Afbd. 


FORMULAS   FOR  WORKING   LOADS. 


59 


54.  Flexure   Formulas   for   Working   Loads   Based    on 
Linear  Variation  of  the  Compression  and  Neglecting  Ten- 
sion in  the  Concrete.  —  The  loads  being  working  loads,  the  unit 
stress  in  the  steel  is  within  the  elastic  limit,  and  the  unit  stresses 
in  the  concrete  vary  as  the  ordinates  to  the  compressive  stress- 
strain  curve  for  concrete  up  to  working  stresses.    This  curve 
is  nearly  straight;   it  will  be  assumed  straight  to  simplify  the 
formulas.     The  resulting  errors  are  small,  as  is  explained  in 
Art.  70. 

55.  Neutral  Axis  and  Arm  of  Resisting  Couple.  —  It  follows 
from  the  assumption  of  plane  sections  that  the  unit  deformations 
of  the  fibres  vary  as  their  dis- 

tances from  the  neutral  axis; 
hence,  es/ec=  (d-kd)/kd  (see 
Fig.  16).  Also  es  =  fs/Es  and 
ec  =  fc/Ec;  hence,  introducing 
the  abbreviation  n, 

f8     d-kd    l-k 

~  "  k    '  •'  (a) 


c~    kd 

When  the  loads  and  reactions 
are  vertical  —  beam  horizontal 
—the  total  tension  and  compression  on  .the  section  are  equal,  i.e., 

.fiA-lfokd  ......     .     .  '(6) 

Eliminating  /,//c  between  equations  (a)  and  (b)  and  introduc- 
ing the  abbreviation  p  gives  2pn(l-k)  =k2;  this  if  solved  for  k 
gives 

pn  ......     (1) 


FIG.  16. 


This  formula  shows  that  the  neutral  axes  of  all  beams  of  a 
given  concrete  and  of  a  given  percentage  of  reinforcement  are 
at  the  same  proportionate  depth,  k,  for  all  working  loads.  The 
lower  group  of  curves  in  Fig.  17  gives  k  for  different  values  of 
p  and  n;  thus  tor  p  =0.015  (percentage  of  steel  =  1.5)  and 
n  =  15,  k  =0.48.  The  curves  show  that  k  increases  as  p  or  n 


60  GENERAL  THEORY.  [Cn.  Ill 

The  distance  of  the  centroid  of  the  compressive  stress  from 
the  compressive  face  of  the  beam  is  %kd;  therefore  the  arm  of 
the  resisting  couple,  TC,  is  given  by  ' 

jd=d-\kd,    or     /  =  1-J&  .....     (2) 

As  k  increases,  /  decreases,  but  not  in  the  same  ratio.  Fig.  17 
shows  how  j  changes  with  p  for  four  different  values  of  n.  It 
should  be  noticed  that  /  does  not  vary  much  with  p,  and  that 
for  n  =  15  and  p  between  0.75  and  1.0%  —  common  values— 
the  average  value  of  /  is  about  J. 

56.  Resisting  Moment  for  Given  Working  Stresses  /„  and  fc.  —  • 
If  the  beam  is  under-reinforced,  its  resisting  moment  depends 

on  the  steel  and  its  value  then  is  ^  x;  n- 

\ 

:>  .     .     .     .     (3) 


• 

If  over-reinforced,  the  resisting  moment  depends/ni,  the  concrete 
and  its  value  then  is 

irrrlfy* 

Mc  =  C-jd=$fcbkd'jd  =  %fckjbd2.    .  •  .    .    .     (4) 

To  find  the  resisting  moment  in  a  given  case,  these  values  of 
M  must  be  compared,  and  the  lesser  one  taker  ;  v>ut  it  may  be 
noticed  that  a  comparison  of  the  quantities  fsp  and  %Uc  is 
sufficient  to  determine  which  of  the  values  is  the  lesser. 

For  approximate  computations  one  may  use  the  average 
values  /=J  and  &  =  f;  then  formulas  (3)  and  (4)  become 
respectively 

Af.=M-K    ...'....    (3)' 

(4)' 


57.  Unit  Fibre  Stresses  /or  a  Given  Bending  Moment.  — 
Formulas  for  these  may  be  obtained  from  equations  (3)  and  (4) 
by  solving  them  for  fs  and  /«.  respectively;  M  will  denote  bend- 
ing moment.  Or,  one  may  reason  as  follows:  Since  the  resisting 
moijnent  is  Tjd, 

M  T 

T=         and    /.  =     ;   ......     (5) 


§57.] 


FORMULAS  FOR   WORKING   LOADS. 


61 


i.oo 


A 


V 


-40- 


05^ 


Pjercejitage  of  Ste^l 


FIG.  17. 


62  GENERAL  THEORY. 

also,  since  fc  equals  twice  the  average  unit  compressive  stress 
on  the  section,  and  C=T, 


Approximating  as  before,  i.e.,  using  average  values  /—  J  and 
%,  formulas  (5)  and  (6)  become  respectively 

M  T 

r-        and    /.-,  ......    (5)' 


2T 
and  /•--W'P-      .......     (6)' 


58.  Determination  of  Amount  of  Steel  and  Cross-section 
of  Beam  for  a  Given  Bending  Moment. — If  k  be  eliminated  be- 
tween equations  (a)  and  (6),  the  following  formula  for  steel 
ratio  results: 

-          V2  (7) 


It  shows  that  for  given  concrete  and  ratio  of  working  stresses, 
p  has  the  same  value  for  all  .sizes  of  beams.  Fig.  18  gives 
graphically  the  proper  values  of  p  for  different  ratios  }8/fe 
and  four  different  values  of  n. 

If  a  value  of  p  less  than  that  given  by  (7)  is  adopted  then 
the  cross-section,  or  bd2  rather,  should  be  determined  from  the 
first  of  equations  (8)-,  if  greater,  from  the  second.  (These  are 
(3)  and  (4)  solved  for  bd2  respectively.) 


f*P1 


Values  of  k  and  j  can  be  obtained  from  (1)  and  (2)  or  Fig.  17; 
then  inserting  an  assumed  value  of  b,  d  can  be  obtained  by 
direct  solution  of  the  formula. 


59.] 


FORMULAS  FOR  WORKING  LOADS. 


63 


For  Approximate  Design. — To  determine  the  percentage  of 
steel,  use  (6)'  in  this  form,  p  =  &/<•//«.  If  a  smaller  percentage 
than  this  is  decided  upon,  use  the  first  of  equations  (8)'  to 
determine  b  and  d\  and  if  a  larger  then  the  second  one. 


M 


(8)' 


59.  Diagrams   and   Examples. — Some   numerical    examples 
illustrating  the  preceding  principles  will  now  be  given,  and 


FIG.  18.  ~ 

then  some  diagrams  will  be  explained  by  means  of  which  com- 
putations in  such  examples  can  be  wholly  avoided  or  nearly  so. 

(1)  A  concrete  beam  is  10  X 16  inches  in  cross-section  and  the  tension 
reinforcement  consists  of  four  f  inch  steel  rods,  their  centres  being  two 
inches  above  the  lower  face  of  the  beam.  The  working  stress  of  the  con- 
crete being  600  Ibs/iri2  and  that  of  the  steel  15,000,  what  is  the  safe 
resisting  moment  of  the  beam? 

Solutions.  The  cross-section  of  one  steel  rod  is  0.442  in2,  hence 
A  =  1.768;  and  as  6=10  and  d=14;  p  =  1.768/140  =  0.0126.  There- 
fore, n  being  taken  as  15,  from  (1)  k  =  0.453;  also  from  (2)  /= 0.849. 
As  determined  by  the  steel,  the  resisting  moment  is  (see  eq.  3) 

3/8=15.000x1.768x0.849x14  =  315,000  in4 


64  GENERAL  THEORY.  [Cn  III, 

As  determined  by  the  concrete,  the  resisting  moment  is  (see  eq.  4) 
Me  =  300  X  10  X  0.453  X  14  X  0.849  X  14  =  227,000  in-lbs. 

The  safe  resisting  moment  is  the  latter  value. 

The  approximate  formulas,  (3)'  and  (4)',  give  respectively 

M8  =  15,000  XI.  768X|X14  =  325,000 
and  M  c  =  600  X  i  X  10  X  142  =  196,000  in-lbs. 

The  approximate  formula  relating  to  the  steel  always  gives  a  closer 
result  than  the  other. 

(2)  Suppose  that  the  beam  of  the  preceding  example  is  19  in.  deep 
and  is  subjected  to  a  bending  moment  of  350,000  in-lbs.      Compute  the 
greatest  unit  stresses  in  the  steel  and  concrete. 

Solutions.  The  steel  ratio  is  1.768/170  =  0.0104;  and  with  n=15, 
eq.  (1)  gives  A:  =  0.424,  and  eq.  (3)  gives  /=  0.859.  Therefore 
T  =  350,000/0.859  X  17  =  24,000  Ibs.,  and  /.  =  24,000/1.768  =  13,600  lbs/in3. 
Also  see  eq.  (6),  /c  =  48,000/0.424x10x17  =  665  lbs/in2. 

The  approximate  formulas  (5)'  and  (6)'  give  respectively 

/.  =  13,500    and    fc  =  750  lbs/in2. 

Again,  of  the  approximate  formulas,  the  one  relating  to  the  steel  gives 
the  closer  result. 

(3)  A  beam  is  to  be  figured  to  withstand  a  bending    moment  of 
135,000  in-lbs.,  the  working  strength  of  the  concrete  and    steel  being 
taken  at  700  and  12,000  lbs/in2  respectively. 

Solutions.  For  ft  =  15,  eq.  (7)  gives  p  =  0.0136.  With  this  value  of 
p,  eq.  (1)  gives  A:  =  0.462,  and  hence  /  =  0.846.  Eq.  (8)  now  gives 

M»  =  135,000 

12,000X0.0136X0.846 

Many  different  values  of  b  and  d  will  satisfy  the  last  equation.    If  6  is 
taken  as  7  in.,  then 

140      or    d=12  in. 


Finally  A  =  0.0136(7X12)  =  1.14  in2. 

The  approximate  formula  6'  gives  for  a  suitable  steel  ratio  p=& 
700/12,000  =  0.0109.  Adopting  0.01  1,  then  8'  gives  bd*  =  135,000  /£  700  = 
1157.  Taking  6  =  7  in.  as  before,  d*=  1157/7  =  165.3,  or  d  =  12.8,  13.  in. 
say.  Finally  A  =0.011x7X13  =  1.00  in2. 

The  construction  of  the  diagram-  (Plates  I-IV,  pages  275  to 
278)  referred  to  will  now  be  explained  and  then  their  use.  It  will 
be  convenient  to  have  names  for  the  quantities  fspj  and 


§59]  FORMULAS  FOR  WORKING  LOADS.  65 

(see  eqs.  3  and  4)  and  single  symbols  for  them.  We  shall  call 
them  coefficients  of  resistance  relative  to  the  steel  and  the  concrete 
and  will  denote  them  by  Rs  and  Rc  respectively;  that  is, 

(a)     R8=fsPJ    and     (b)    Rc  =  %fckj. 

Then  the  formulas  for  resisting  moments  of  a  given  beam 
with  particular  working  strengths  fs  and  fc  may  be  written  thus  : 

Ms  =  Rsbd2    and    Me=Rcbd2  .....     (1) 

Similarly  for  any  particular  beam  subjected  to  a  bending 
moment  M, 

(2) 


Likewise   for  any  particular  bending   moment   and  working 
strengths  f8  and  }C)  the  necessary  section  is  given  by 


(3) 


R  being  the  smaller  of  the  two  coefficients  of  resistance. 

In  the  four  diagrams  values  of  p  are  given  at  the  upper 
and  lower  margins  and  values  of  R8  and  Rc  at  the  sides.  The 
diagrams  are  drawn  for  four  different  values  of  n,  viz.,  10,  12, 
15,  and  18,  as  noted  on  the  plates. 

The  f8  curves  of  the  diagrams  are  merely  the  plots,  or 
graphs,  of  equation  (a)  for  certain  values  of  /«  as  marked  on 
the  curves.  The  fc  curves  are  the  graphs  of  equation  (b)  for 
various  values  of  fc  as  marked.  For  example,  when  n=15, 
/.  =  14,000,  /c  =  600,  and  p  =  l%  (see  page-*  277),  R8=120  and 


The  foregoing  three  examples  will  now  be  solved  by  means 
of  the  diagram,  page  277  (n  =  15). 

(1)  The  percentage  of  steel  being  1.26,  we  first  find  that  value  on 
the  lower  margin;  then  trace  vertically,  stopping  at  the  first  of  the  two 
curves  fc  =  600  and  /8  =  15,000;  then  trace  horizontally  to  either  side 

*  These  diagrams  are  modeled  after  those  contributed  by  Prof.  French  in 
Trans,  Am.  Soc.  C.  E..  Vol.  LVI,  1906,  pp.  362-4. 


68 


GENERAL  THEORY. 


[Cn.  III. 


margin  and  read  off  the  value  72  =  115.     Finally  M  =  115X10  Xl42  = 
225,400  in-lbs. 

(2)  R  =  M/bd2  =  350,000/10  -172  =  121,  and  the  percentage  of  steel  is 
1.04.    We  enter  the  diagram  with  these  values  of  R  and  p,  find  the  inter- 
section of  the  horizontal  and  vertical  lines  through  these  values  respect- 
ively, and  from  the  steel  and   concrete  curves  adjacent  to  this  inter- 
section estimate  /8  to  be  13,750  and  fc  675  lbs/in2. 

(3)  We  first  find  the  intersection  of  the  curves  /c  =  700  and /s=  12,000; 
from  that  point  tracing  down  we  find  p  =  1.35%,  and  tracing  horizontally 
we  find  R  =  137.    Then  bd2  =M/R  =  135,000/137  =  986,  from  which  b  and 
d  may  be  decided  upon,  and  then  finally  the  amount  of  steel. 

60.  Flexure  Formulas  for  Ultimate  Loads,  Based  on 
Parabolic  Variation  of  Compression  and  Neglecting  Tension 
in  Concrete. — It  is  assumed  that  the  amount  of  reinforce- 
ment is  sufficient  to  develop  the  full  compressive  strength 
of  the  concrete  without  straining  the  steel  beyond  its  yield 
point;  or  otherwise  expressed,  failure  occurs  by  crushing  of 
the  concrete,  the  stress  in  the  steel  being  still  within  the 
yield  point.  Then  the  parabola  representing  the  variation  of 
compression  is  a  full  parabola  (see  Art.  26),  the  upper  end 
(see  Fig.  19)  being  the  vertex. 


FIG.  19. 

If  the  amount  of  steel  in  a  beam  is  such  that  the  ultimate 
strength  of  the  concrete  and  the  elastic  limit  of  the  steel  would 
be  reached  simultaneously  if  the  beam  were  subjected  to  a  gradu- 
ally increasing  loa'd,  then  this  will  be  called  the  ideal  amount — 
no  better  term  seems  available — but  this  amount  may  not  be 
the  best  in  a  given  case. ) 


§61.]  FORMULAS  FOR   ULTIMATE  LOADS.  67 

In  the  present  connection,  the  two  following  properties  of 
a  parabola  like  that  of  Fig.  19  are  useful:  (1)  The  average 
abscissa  of  the  parabolic  arc  equals  two-thirds  the  greatest,  /c; 
(2)  the  distance  from  the  centroid  of  the  parabolic  area  to 
its  top  equals  three-eighths  the  total  height,  kd. 
->  6r.  Neutral  Axis  and  Arm  of  Resisting  Couple.—  The  "initial 
modulus  of  elasticity"  of  the  concrete  (Art.  24)  is  denoted  by 
Ec  in  the  present  article.  It  is  represented  by  the  tangent 
of  the  angle  between  the  vertical  through  N  and  the  tangent 
to  the  stress-strain  curve  at  N.  And  since  NA  represents  ec, 
it  follows  from  a  well-known  property  of  the  parabola  that 
fc=%Ecec.  Also  fs  =  Eses,  and  from  the  assumption  of  plane 
sections  it  follows  that  e8/ec=(d—kd)/kd.  Eliminating  es/ee 
from  the  above  equations,  and  introducing  the  abbreviation  n, 
gives 

f*   l~k 


When  the  loads  and  reactions  are  vertical  —  beam  horizontal  — 
the  total  tension  and  the  total  compression  on  the  section 
are  equal,  i.e., 

f8A  =  zfcbkd  .......    .    (b) 

Eliminating  f8/fc  between  equations  (a)  and  (6)  and  intro- 
ducing the  abbreviation  p,  gives  3pn  =  k2/(l  —  k);  this  if  solved 
for  k  gives 

.     (1) 


This  formula  shows  that  the  neutral  axes  of  all  beams  of  a 
given  concrete  and  of  a  given  percentage  of  reinforcement 
are  at  the  same  proportionate  depth,  k,  for  their  respective 
ultimate  loads.  The  lower  group  of  curves  (Fig.  20)  gives  k 
for  different  values  of  p  and  n;  thus  for  p  =  2%  and  n  =  15, 
k  =  0.60.  The  curves  show  that  k  increases  as  p  or  n  increases. 
The  distance  of  the  centroid  of  the  compressive  stress  from 
the  compressive  face  of  the  beam  is  pd;  therefore  the  arm 
of  the  resisting  couple  TC  is  given  by 

jd=d-%kd,    or    /  =  l-p  .....    (2) 


63 


GENERAL  THEORY. 


[On.  IH 


FIG.  20. 


§63.]  FORMULAS  FOR  ULTIMATE  LOADS.  69 

Plainly,  as  k  increases  /  decreases,  but  not  at  the  same  rate. 
The  upper  group  of  curves  in  Fig.  20  gives  /  for  different  values 
of  p  and  n;  thus  for  p  =  2%  and  n  =  15,  j  =0.775.  It  should 
be  noticed  that  /  does  not  vary  much  with  p,  and  that  for 
n  =  15  and  p  greater  than  1%  the  average  value  of  /  is  about 
0.80. 

4  62.  Ultimate  Resisting  Moment  for  a  Given  Ultimate 
Strength  fc.  —  Remembering  the  assumption  made  at  the  outset 
in  regard  to  the  amount  of  steel  (Art.  60),  it  will  be  under- 
stood that  the  ultimate  resisting  moment  always  depends  on 
the  concrete;  the  va'ue  is 

Mc  =  C-jd  =  %fcbkd.jd=%jkf(l)d2  .....     (3) 

It  should  be  remembered  that  this  equation  gives  the  ultimate 
resisting  moment  only  if  when  the  unit  stress  in  the  concrete 
is  at  the  ultimate  that  in  the  steel  is  not  beyond  the  elastic 
limit. 

If  the  beam  has  the  "  ideal  amount  "  of  reinforcement 
before  referred  to,  then  the  ultimate  resisting  moment  can  be 
computed  from  the  steel  by  means  of 

Ms  =  T-jd=f8A.jd=f8pjbd*,      ....     (4) 

in  which  /,  denotes  elastic  limit  of  steel. 

For  approximate  computations  one  may  use  the  average 
values  /  =  0.80  and  &=0.52;  with  these,  formulas  (3)  and  (4) 
become  respectively 

Mc=Q.278fcbd2,      .......     (3)' 

.     .....     (4)' 


63.  Determination  of  Amount  of  Steel  and  Cross-section  of 
Beam  for  a  Given  Ultimate  Bending  Moment.  —  When  a  beam 
contains  the  "ideal  amount"  of  steel,  the  values  of  M  given 
by  (3)  and  (4)  are  equal;  hence,  fs/fc  =  2k/3p.  If  the  value 


70 


GENERAL  THEORY. 


[Cn.  III. 


of  k  as    given  by  equation  (a)  be  inserted  in  this  equation, 
then  the  following  formula  for  the  "ideal  steel  ratio'7  results: 


2/3 


(5) 


This  shows  that  p  depends  only  on  the  ultimate  strength  of 
concrete  and  elastic  limit  of  steel,  and  not  at  all  on  the  size 
of  beam.  Fig.  21  gives  graphically  the  "ideal  ratio"  p  for 


50 


40- 


10- 


Perce 


itage  of  Stee 


. 


FIG.  21. 

different  values  of  the  ratio  f8/fe  and  four  values  of  n;    thus 
for  /8  =  34,000,  /c  =  1700,  and  n  =  15,  p  =  1.93%. 

If,  in  any  given  case,  the  steel  ratio  as  given  by  (5),  or  a 
higher  value,  is  adopted,  then  the  concrete  would  crush  without 
straining  the  steel  beyond  the  elastic  limit,  and  the  ultimate 
resisting  moment  of  the  beam  is  given  by  (3),  which  value 
equated  to  the  ultimate  bending  moment,  M,  to  be  provided 
for,  gives  %fcjkbd2  =  M,  or 


§64.]  FORMULAS  FOR  ULTIMATE  LOADS  71 

From  this  d  may  be  computed  for  any  assumed  value  of  b. 
If  a  lower  value  than  that  given  by  equation  (5)  is  adopted 
for  p,  then  under  a  gradually  increasing  load  the  stress  in 
the  steel  would  reach  the  elastic  limit  before  the  concrete 
would  crush,  and  the  formulas  of  this  article  could  not  be 
used  to  compute  the  ultimate  resisting  moment  of  the  beam. 
See  Art.  67  for  solution  of  this  case. 

Approximating  as  before,  j=0.80  and  k  =  0.52,  and  eq.  (6) 
becomes 


(6y 


64.  Diagrams  and  Examples.  —  Two  numerical  examples 
will  now  be  given  to  illustrate  the  foregoing  principles,  and 
then  a  diagram  will  be  explained  by  means  of  which  compu- 
tations in  such  examples  can  be  wholly  or  partially  avoided. 

(1)  A  concrete  beam  is  10x16  inches  in  cross-section  and  the  tension 
reinforcement  consists  of  four  f-in.  steel  rods,  their  centers  being  two 
inches  above  the  lower  face  of  the  beam.  The  ultimate  compressive 
strength  of  the  concrete  being  2000  and  the  elastic  limit  of  the  steel 
40,000  lbs/in2  compute  the  ultimate  resisting  moment  of  the  beam. 

Solutions.  Here  p  =  0.0126,  and  for  n  =  15,  eq.  (1)  gives  7c  =  0.52 
and  (2)  gives  /  =  0  .  805.  Hence 

Mc  =  %  0.805  X  0.52  X  2000  X  10  X142  =  1,096,000  in-lbs. 

It  remains  to  test  whether  the  stress  in  the  steel  would  be  within  the 
elastic  limit,  the  beam  being  subjected  to  a  bending  moment  of  1,096,000 
in-lbs.  This  is  done  by  dividing  the  bending  moment  by  the  arm  of  the 
resisting  couple,  which  gives  the  whole  tension  in  the  steel,  and  then 
this  tension  by  the  area  of  the  steel;  thus 


0.805  X 14 
and 


-55,000  lbs./in>=/8 

This  result  being  beyond  the  stated  elastic  limit,  eq.  (3)  does  not  apply 
to  the  problem  in  hand.  (The  ultimate  resisting  moment  can  be  com- 
puted by  other  methods.  See  ex.  2,  page  80.) 


72  GENERAL   THEORY.  [Cn.  III. 

(2)  A  beam  is  to  be  figured  to  safely  withstand  a  bending  moment 
of  135,000  in-lbs.,  the  ultimate  compressive  strength  of  the  concrete 
being  taken  at  2000  and  the  elastic  limit  of  the  steel  at  40,000  lbs/in2. 

Solution.  With  n=15,  eq.  (5)  gives  as  the  "  ideal  steel  ratio,"  since 
A//c=20, 

2/3  •••'• 


For  this  value  of  p,  eq.  (1)  gives  k  =  0.598,  and  (2)  gives  /=  0.775.  With 
a  factor  of  safety  of  3,  the  ultimate  bending  moment  is  405,000  in-lbs  ., 
and  eq.  (6)  gives 

405  000 
^2==fX2000X0.775X0.598==656  ^ 

Trying  6  inches  for  6,  then  d2  =  109.3  or  d  =  10.5  in.  ;  also  A  =  0.02  X  6  X  10.5 
=  1.26  in2. 

The  "  coefficients  of  resistance."  on  the  parabolic  theory 
are  }9pj  and  $fejk  (see  equations  4  and  3),  and  using  the  sym- 
bols Rs  and  Rc,  as  in  Art.  59, 

Rs=f8pj    and    Re  =  %fjk. 

The  /,  curves  of  the  diagram  (Plate  V,  page  279)  are  graphs  of  the 
first  equation  for  certain  values  of  }8  as  marked  on  the  curves 
and  n  =  15.  (The  curves  for  n  =  12  differ  very  little  from  these  .) 
The  fc  curves  are  graphs  of  the  second  equation  for  various 
values  of  fe  as  marked;  the  full  curves  are  for  n  =  15  and  the 
dotted  for  n  =  12. 

In  using  the  diagram  to  determine  (1)  the  ultimate 
resisting  moment  of  a  given  beam  for  a  specified  ultimate 
compressive  strength  of  the  concrete,  or  (2)  a  steel  ratio  and 
size  of  beam  to  withstand  a  given  ultimate  bending  moment 
with  specified  compressive  strength  of  concrete,  these  formulas 
respectively  should  be  borne  in  mind: 

M  =  Rbd2    and     bd2 


The  foregoing  two  examples  will  now  be  solved  by  means  of 
the  diagram. 

(1)  The  percentage  of  steel  being  1.26,  we  first  find  that  value  on 
the  lower  margin  of  the  diagram,  and  then  trace  vertically  to  the  line 
marked  fe  —  2000.  We  note  that  the  point  thus  found  is  above  the  line 


65.] 


GENERAL  PARABOLIC  FORMULAS. 


73 


f8  =  40,000,  the  elastic  limit  of  the  steel  of  the  beam,  and  hence  conclude 
that  the  amount  of  steel  in  this  beam  is  insufficient  to  develop  the 
full  compressive  strength  of  the  concrete  without  straining  the  steel 
beyond  the  elastic  limit.  If  the  elastic  limit  of  the  steel  were  as  high 
as  55,000  lbs/in2,  we  would  trace  horizontally  from  the  point  as 
found  above  to  either  side  of  the  diagram  and  read  #  =  552.  Then 
M=Rbd2  =  552  X 10  X 142  =  1,087,000  in-lbs.,  which  is  the  ultimate  resist- 
ing moment  of  this  beam  with  the  high  elastic  limit  steel. 

(2)  We  first  find  the  intersection  of  the  curves  /c  =  2000  and 
}8  =  40,000 ;  from  that  point  tracing  down  we  find  p  =  2%,  and  horizontally 
we  find  #  =  620.  Then  M*  =  M/R  =  405,000/620  =  654,  from  which  b 
and  d  can  be  decided  upon,  and  finally  the  amount  of  steel. 


65.  Flexure  Formulas  for  any  Load  up  to  Ultimate, 
Based  on  Parabolic  Variation  of  Compression  and  Neglect- 
ing Tension  in  Concrete  (After  Talbot). — It  is  assumed  that 
the  stress  in  the  steel  is  not  above  the  yield  point.  The  par- 
abola representing  the  variation  of  compressive  stress  is  not  a 
"full  one",  that  is,  its  top  is  not  the  vertex,  see  Fig.  22,  unless 

//"I 

f,  ;/. 


kd 


the  maximum  concrete  stress  is  at  the  ultimate  value.  As 
heretofore  fc  and  ec  will  denote  the  unit  stress  and  strain  re- 
spectively at  the  compressive  face  of  the  concrete,  and  as  in 
Art.  61,  EC  will  denote  the  initial*  modulus  of  elasticity  of  the 
concrete.  In  this  article  //  and  ecf  will  denote  these  same 
quantities  at  the  ultimate  stage  of  the  concrete,  and  q  will 
be  used  as  an  abbreviation  for  ec/ec' '.  It  can  be  shown  from 


74 


GENERAL  THEORY. 


[Cn.  III. 


the 

tot 

1.0 
0.9 
0.8 
0.7 

°0.5 
|0.4 
0.3 
0.2 
0.1 

R 

properties  of  a  parabola  that:    (1)  The  average  abscissa 
he  parabola  NB  is  (3  —  q)/3(2  -q)  times  the  greatest  abscissa 

\ 

X 

X 

X 

X 

1 

^x 

X 

y 

/ 

X 

^ 

/ 

X 

^ 

^ 

/ 

s 

~7 

/ 

X 

x 

X^ 

X 

^ 

f 

X 

^ 

1 

j 

.1     .2     .3      .4     .5     .6     .7      .8     .9     1.0 
Values  of /^/c' 

PIG.  23a. 


235. 


}c;    (2)  the  distance  from  the  centroid  of  the  parabolic  area 
to  the  top  AB  is  (4-g)/4(3  —  q)  times  its  height,  kd;  and  (3) 


§  66.]       FLEXURE  FORMULAS  AFTER  TALBOT.         75 

Fig-.  23a  shows  graphically  the  relation  between  q  and  the 
ratio  fc/fc'j  thus  when  q  =  l  (the  concrete  is  strained  to  one- 
fourth  its  limit  of  compression)  the  unit  stress  in  the  concrete 
is  about  0.45  of  the  ultimate  strength. 

The  lines  NB  in  Fig.  236  show  the  distributions  of  com- 
pressive  stress  at  a  section  of  a  beam  when  q  is  },  J,  f  and  1 
respectively  as  marked.  In  each  case  N  is  the  neutral  axis 
and  AB  represents  the  unit  stress  on  the  remotest  fiber.  When 
q  is  J,  the  distribution  is  almost  linear. 

66.  Neutral  Axis  and  Arm  of  Resisting  Couple.  —  As  in 
Arts.  55  and  61,  e8/ec  =  (d—kd)/kd,  and  }8=E8e8.  Eliminating 
es/ec  from  these  two  equations  and  (a),  and  introducing  the 
abbreviation  n,  gives 


When  the  loads  and  reactions  are  vertical—  beam  horizontal  — 
the  total  tension  and  total  compression  on  the  section  are 
equal,  i.e., 

Af.=bkdfc(3-q)/3(2-q).    ....     .     (c) 

Eliminating  the  ratio  fs/fc  between  equations  (6)  and  (c),  and 
introducing  the  abbreviation  p,  gives  6pn(l—  k)  =&2(3—  q), 
which  solved  for  k  furnishes  the  following  formula: 


It  shows  that  the  neutral  axes  of  all  beams  of  a  given  con- 
crete and  a  given  percentage  of  reinforcement  are  at  the  same 
proportionate  depth,  k,  for  any  particular  stage  of  loading 
as  given  by  q.  The  lower  group  of  curves  in  Fig.  24  shows 
how  k  depends  on  p  and  n  for  q  =  J,  the  value  taken  by  Talbot 
as  closely  corresponding  to  the  working  stage.  The  lower 


76 


GENERAL  THEORY. 


[Cn.  IH. 


FIG.  24. 


§  68.]  FLEXURE  FORMULAS  AFTER  TALBOT.  77 

group  of  curves  in  Fig.  25  shows  how  k  depends  on  q  (that  is, 
on  the  stage  of  loading)  for  several  values  of  p,  n  being  taken 
as  15.  Thus  when  p  =  0.01  and  q^Q  nearly  (load  very  small), 
k  =  0.42:  and  when  q  =  l  nearly  (ultimate  load),  k  =0.48. 

The  distance  of  the  centroid  of  the  compressive  stress  from 
the  toj)  of  the  beam  is  &d(4-g)/4(3-g);  hence  the  arm  of 
the  resisting  couple  is  given  by  jd=d—  kd(4-q)/4(3  —  q)  or 


4(3-qY  '    •    •:••.••    ' 

The  upper  group  of  curves  in  Fig.  24  shows  how  /  depends 
on  p  and  n,  for  the  stage  q  =  \.  The  upper  group  of  curves 
in  Fig.  25  shows  how  /  depends  on  q  for  several  values  of  p, 
n  being  taken  as  15.  It  should  be  noticed  that  /  does  not 
change  much  for  considerable  changes  in  q. 

67.  Resisting  Moment  for  Given  Values  of  fc  and  fs.  —  Whether 
the  resisting  moment  is  determined  by  the  concrete  or  steel 
depends  on  the  percentage  of  reinforcement;  in  a  general 
way  the  higher  percentages  make  the  moment  depend  on 
the  concrete"  and  the  lower  on  the  steel.  As  depending  on 
the  concrete,  the  resisting  moment  is  given  by 


(3) 


The  value  of  q  to  be  used  ^  here  must  correspond  with  the  fc 
used,  the  relation  between  q  and  fc  being  given  by  (a)  of  Art.  65; 
or  by  Fig.  23a.  As  depending  on  the  steel,  the  resisting  mo- 

ment is 

(4) 


68.  Determination   of  Fibre  Stresses  f8  and  fc  for  a  Given 
Bending  Moment.  —  Formulas  for   these   can  be  obtained  by 
solving  (3)  and  (4)  for  }c  and  f8  respectively;  thus 
,     3(2-  q)    M 


M 
pjbd* 


78 


GENERAL  THEORY. 


[CH,  in. 


1.00 


.'JO 


.70 


The  curves  give  values  of  J  and  k  according 
to  the  theory  of  Talbot;  the  heavy  horizontal  lines 
give  values  according  to  theory  based  on  the 
lineal  law  of  distribution  of  the  compressive 
fiber  stress. 


All  Values  of 


j  and  k  ate  based  on  n 


Values  of    Q. 


-15. 


.40 


.50 


0)    -H 

Q  g 


,70 


.so 


.90 


1.00 


FIG.  25. 


§69.]  FLEXURE  FORMULAS  AFTER  TALBOT.  79. 

Neither  fc  nor  fs  can  be  determined  directly  from  these,  for  each 
formula  contains  q  (j  and  k  depend  on  q),  which  is  an  unknown 
in  the  problem  in  hand.  An  estimated  value  of  q  must  be  used 
for  a  trial  solution  of  (5),  and  then  with  the  value  of  fc  thus 
found  a  better  value  of  q  may  be  obtained  from  (a)  or  from 
Fig.  23a,  which  value  may  be  used  in  a  second  trial  solution. 

69.  Determination  of  Amount  of  Steel  and  Cross-section  of 
Beam  fora  Given  Bending  Moment.  —  In  order  that  the  maximum 
unit  compression  in  the  concrete,  fc,  and  the  unit  stress  in  the 
steel,  f8,  may  have  certain  definite  values  when  the  beam  is 
subjected  to  a  given  bending  moment,  a  certain  definite  per- 
centage of  steel  must  be  used.  This  percentage  is  such  as 
makes  the  values  of  the  resisting  moment  as  determined  by 
steel  and  concrete  equal.  Thus  equating  values  of  M  from 
equations  (3)  and  (4)  and  simplifying, 


Inserting  in  this  the  value  of  k  furnished  by  (6)  gives 

3-g  1 

3(2-9)  /./2- 


In  this  also  the  value  of  q  used  should  correspond  to  the  value 
of  fc  adopted  as  working  stress.  The  curves  of  Fig.  26  give 
values  of  p  for  different  values  of  fs/fc  up  to  50,  q  being  taken 
at  1. 

If  in  any  given  case  a  value  for  p  less  than  that  given  by 
(6)  is  adopted,  then  the  resisting  moment  is  given  by  equa- 
tion (4),  which  equated  to  the  bending  moment  to  be  provided 
for  gives  fspjbd2  =  M,  or 

M»-^..  (7) 

fsP] 

If  a  greater  value  of  p  is  adopted,  then  the  resisting  moment 


80  GENERAL  THEORY.  [Cn.  III. 

is  given  by  (3),  which  if  equated  to  the  bending  moment  gives 


bd2 


3(2-  q)  M_ 
3-g    jkfe' 


(8) 


From  the  proper  one  of  these,  d  can  be  computed  for  any 
assumed  value  of  b. 


50 


^ 


40- 


X 


»s 
1 


0,5* 


Percentage 

W 


of.S 


eel 


1.5* 


FIG.  26. 

Examples. — (1)  It  is  required  to  solve  example  1,  Art.  59,  by  the 
methods  of  this  article,  it  being  supposed  that  for  the  working  stress 
/c  =  600  lbs/in2,  q  =  \. 

Solution.  As  shown  in  the  solution  of  the  example  referred  to, 
A  =  1.768  in2,  and  p  =  0.0126;  therefore  from  eq.  (1)  or  Fig.  24,  n  being 
taken  as  15,  k  =  0.466,  and  from  eq.  (2)  or  Fig.  24,  /= 0.842.  Then  from 
eqs.  (3)  and  (4) 

Mc  =  l\  X 0.842  X 0.466  X  600  X 10  X 142  =  242,000  in-lbs. 
and          Ms  =  15,000  X  1.768  X  0.842  X 14  =  313,000  in-lbs. 

(2)  It  is  required  to  solve  example  1  of  Art.  64  by  the  methods  of 
this  article. 

Solution.  As  disclosed  by  the  solution  in  Art.  64,  the  stress  in 
the  steel  will  reach  the  elastic  limit  before  that  in  the  concrete  would 


§70.]       FLEXURE  FORMULAS  AFTER  TALBOT.        81 

reach  the  ultimate  strength;  hence  the  ultimate  resisting  moment  de- 
pends on  the  steel.  The  stress  existing  in  the  concrete  when  the  steel 
is  stressed  to  the  elastic  limit  is  unknown ;  so  is  q.  Supposing  that  this 
stress  in  concrete  is  f  the  ultimate  strength,  <?  =  0.5  (see  Fig.23a) ;  then, 
since  p  =  0.0126,  and  n  is  taken  as  15,  fc  =  0.48  and  ;  =  0.83  (see  Fig.  25), 
and  eq.  (4)  gives  M8  =  820,000  in-lbs.  For  a  bending  moment  of  this 
value,  the  stress  in  the  concrete  would  be  (with  the  above  values  of  q,  /, 
and  k)  1260  lbs/in2  (see  eq.  5).  Now  for  the  ratio  1260/2000,  q  is 
about  0.4,  k,  0.75,  and  /,  0.825.  Since  this  value  of  /  is  practically  like 
the  one  used  in  the  trial  computation,  the  ultimate  resisting  moment  may 
be  taken  as  820,000  in-lbs. 

(3)  It  is  required  to  solve  example  2  of  Art.  59  by  the  methods  of 
this  article,  supposing  the  ultimate  compressive  strength  of  concrete  to 
be  2500  lbs/in2. 

Solution.  This  problem  can  only  be  solved  by  trial  because  it  is 
necessary  to  know  q  at  the  outset,  and  q  depends  on  a  quantity  sought, 
/c.  Supposing  that  the  load  is  about  a  safe  one,  then  q  equals  about 
£.  With  this  value,  n  equal  to  15,  and  p  equal  to  0.0104  (already  found 
on  page  60),  7c  =  0.43,  and  /  =  0.85  (see  Fig.  24).  Then  eq.  (5)  gives 
/c  =  630  lbs/in2.  Now  q  depends  on  the  ratio  of  the  working  stress  in 
the  concrete  to  its  ultimate  strength;  for  the  approximate  value,  630, 
the  ratio  is  0.25,  and  eq.  (a),  or  Fig.  23,  gives  q  =  0.15.  With  this  value 
eq.  (1)  gives  &  =  0.432,  eq.  (2),  /=0.854  (see  also  Fig.  25),  and  eq.  (5), 
/c  =  635  lbs/in2.  This  value  is  so  near  the  first  that  #=0.15  must  be 
practically  correct,  and  /= 0.854  may  be  used  to  determine  the  stress  in 
the  steel.  For  this,  eq.  (5)  gives  /8  =  13,700  lbs/in2. 

(4)  It  is  required  to  solve  example  3  of  Art.  59  by  the  methods  of 
this  article,  the  ultimate  compressive  strength  of  the  concrete  being 
taken  at  2000  lbs/in2. 

Solution.  For  the  ratio  700/2000,  q  is  about  0.2  (see  Fig.  23«.)  With 
n  — 15  eq.  (6)  gives  p  =  0.018 ,  For  this  value  of  p,  we  may  use  either 
(7)  or  (8)  to  compute  the  dimensions  of  the  section.  Choosing  (7)  we 
need  first  a  value  of  /,  which  may  be  obtained  from  (2)  and  (1),  or  closely 
enough  from  Figs.  24  or  25;  the  figures  give  /=0.82,  and  eq.  (7)  gives 
6d2  =  763.  With  6  =  7  (as  in  Art.  59)  d  is  10.5  in. 

70.  Comparison  of  Flexure  Formulas  after  Talbot  with  (1) 
those  for  working  conditions  as  given  in  Art.  54-59,  and  (2) 
those  for  ultimate  conditions  as  given  in  Art.  60-64  : 

(1)  The  heavy  horizontal  lines  of  Fig.  25  give  values  of ;  and 
k,  according  to  the  linear  law  (Art.  54),  and  the  curved  lines 


82 


GENERAL  THEORY. 


[Cn.  III. 


those  after  Talbot.  For  g  =  0.25  and  p  =  0.015,  the  difference 
between  the  two  values  of  j  is  represented  by  ad  and  the  dif- 
ference between  the  two  values  of  k  by  b.  For  all  values  of  q 
up  to  0.25  or  0.30  the  first  difference  is  small,  and  so  the  values 
given  by  the  two  formulas  for  fs  must  be  nearly  the  same.  The 
second  difference  is  larger,  and  the  two  formulas  for  fc  will  not 
agree  so  closely.  An  exact  comparison  will  now  be  made. 
Art.  56  gives  (see  eqs.  3  and  4). 


nd 


(The  primes  are  used  to  distinguish  the  symbols  from  the  cor- 
responding ones  in  the  other  formulas.)  Comparing  these  with 
eqs.  (3)  and  (4),  Art.  62,  one  gets 


_ 
f.~f 

As  already  explained,  q  rarely  exceeds  J  for  working  conditions; 
with  this  value  and  n=15f  the  following  table  gives  the  ratios 
/•'//•  and  fc  /fc  f°r  nve  percentages  of  steel.  For  values  of  q 
less  than  J,  the  ratios  are  nearer  unity;  for  g  =  0,  they  are  all 
unity  and  the  two  sets  of  formulas  are  identical. 


p= 

i%. 

*%. 

1%. 

1.5%. 

2%. 

fs7fs 
fc'/fc 

0.995 
1.092 

0.993 
1.091 

0.991 
1.090 

0.990 
1.088 

0.989 
1.086 

The  unit  stresses  in  the  steel  as  given  by  the  two  formulas  are 
practically  identical.  Any  error  involved  in  the  formulas  for 
//,  based  on  the  linear  law,  is  on  the  side  of  safety. 

(2)  For  loads  which  stress  the  concrete  to  the  ultimate 
limit,  the  stress  parabola  of  Fig.  22  is  full  like  that  of  Fig.  19, 
and  3  =  1.  The  formulas  of  Arts.  65-70  for  this  stage  and  those 
of  Arts.  60-64  are  identical. 

71.  Flexure  Formulas  for  T-beams. — The  following  dis- 
cussion is  based  on  the  linear  law  of  compression,  and  it  neglects 


§72.] 


FORMULAS   FOR    T-BEAMS. 


83 


the  tension  in  the  concrete.     The  following  additional  notation 
is  employed  (see  also  Fig.  27) : 

b  denotes  width  of  flange; 

d      "       effective  depth  of  beam; 

b'      "       width  of  web; 

t       "       thickness  of  flange; 

z       "       depth    of    resultant    compression    below    tcp    of 

flange; 
p      "       steel  ratio,  A/bd. 

It  is  necessary  to  distinguish  two  cases,  namely,   (1)  the 
neutral  axis  is  in  the  flange,  (2)  the  neutral  axis  is  in  the  web. 


h ;*---- 


o«© 


r 


FIG.  27. 


Which  of  the  two  cases  is  at  hand  in  any  particular  compu- 
tation may  not  be  apparent  at  the  outset.  This  may  readily 
be  determined,  however,  by  means  of  certain  formulas  to  be 
explained. 

72.  Case  I.  The  Neutral  Axis  in  the  Flange.  —  All  formulas 
of  Arts.  54-58  (except  approximate  ones)  apply  to  this  case. 
It  should  be  remembered  that  b  of  the  formulas  denotes  flange 
—not  web  —  width,  and  p  (the  steel  ratio)  is  A+bd,  not 
A  +  Vd  (see  Fig.  27). 

Approximate  Formulas.  —  Evidently  the  arm  of  the  resisting 
couple,  CT,  is  always  greater  than  d  —  \t;  hence  the  following 
approximate  formulas  err  on  the  side  of  safety: 


and 


84  GENERAL   THEORY.  [Cn.  III. 

These  give  good  results.  There  are  no  satisfactory  correspond- 
ing formulas  based  on  concrete;  for  determining  the  concrete 
stress  use  the  formulas  for  rectangular  beams.  Usually  the 
concrete  stress  will  be  relatively  small,  as  the  flange  of  a 
T-beam  which  comes  under  Case  I  is  generally  much  stronger 
than  the  steel. 

73.  Case  II.  The  Neutral  Axis  is  in  the  Web. — The  amount 
of  compression  in  the  web  is  commonly  small  compared  with 
that  in  the  flange  and  will  be  neglected  in  the  analysis  of  this 
article.  The  formulas  are  thereby  greatly  simplified  and  the 
resulting  error  is  generally  very  small.  To  provide  for  designs 
in  wrhich  the  web  is  very  large  as  compared  to  the  flange, 
formulas  which  take  account  of  web  compression  are  given 
in  Art.  73a. 

Neutral  Axis  and  Arm  of  Resisting  Couple. — Just  as  in 
Art.  55,  eq.  (a), 


nfc       k   ' 
hence  we  have,  in  terms  of  fs  and  fe, 


(a) 


The    average    unit    compressive    stress    in    the    flange    is 
c+/e[l  —  —  j    =/cM  —  —  j,  and  the  whole  compression  is 

fc(  1-777  3)^-     And  since  the  whole  tension  and  whole  com- 
\       2  tea/ 

pression  on  the  section  are  equal, 


Eliminating  fs/fc  between  equations   (a)  and   (6)  we   get   an 
equation  which  when  solved  for  k  gives 


§73.]  FORMULAS    FOR   T-BEAMS.  85 

Substituting  pbd  for  A  we  also  derive  the  form 


k (3) 


The  arm  of  the  resisting  couple  is  d—  z  (see  Fig.  27).  The 
distance  z  is  equal  to  the  distance  of  the  centroid  of  the  shaded 
trapezoid  from  the  top  of  the  beam,  that  is, 


We  also  have 

jd=d-z,       .    ;   .    ....    (5) 

and,  by  substitution  from  (3)  and  (4)  we  have,  in  terms  of 
t/d  and  p, 


6- 


The  neutral  axis  will  be  at  the  junction  of  web  and  flange 
when  t/d  =  k. 

On  Plate  VI,  p.  280,  are  plotted  curves  for  values  of  k  and  / 
for  various  values  of  p  and  of  the  ratio  t/d.  The  value  of  n 
is  taken  at  15.  This  diagram,  as  well  as  eq.  (6),  shows  that 
j  is  affected  very  little  by  changes  in  the  amount  of  steel. 
The  diagram  also  gives  on  the  right  hand  margin  the  values  of 
fg/fc,  corresponding  to  the  various  values  of  k  as  determined 
from  eq.  (1).  The  curves  for  k  and  j  end  at  points  where 


86 


GENERAL    THEORY. 


[On.  ill. 


k  =  t/d.  They  become  horizontal  at  these  points  and  the 
values  of  k  are  equal  to  those  for  rectangular  beams.  (See 
Fig.  17,  Art.  55). 

Resisting  Moment  and  Working  Stresses. — If  the  beam  is 
under-reinforced,  its  resisting  moment  depends  on  the  steel; 
if  over-reinforced,  on  the  concrete.  These  two  values  of  the 
moment  are  respectively 


(7) 


If  one  is  in  doubt  which  of  these  to  use  when  about 
to  compute  the  resisting  moment  of  a  given  beam  with 
specified  working  stresses,  then  both  values  should  be 
computed  and  the  smaller  taken  as  the  resisting  mo- 
ment. 

The  unit  stresses,  /,  and  fc,  produced  by  a  certain  bending 
moment  M  in  a  given  beam  can  be  computed  by  solving  (7) 
for  fs  and  fc  or  from 


/_£  _A_ 

Jc~n'l-k 


fsP 


II— 1- 

\      2kd/d 


....     (8) 


Approximate  formulas  corresponding  to  (7)  and  (8)  can  be 
established  as  follows:  From  the  stress  diagram  in  Fig.  27, 
it  is  plain  that  the  arm  of  the  resisting  couple  is  never 
as  small  as  d  —  \i,  and  that  the  average  unit  compressive 
stress  is  never  as  small  as  J/c,  except  when  the  neutral 
axis  is  at  the  top  of  the  web.  Using  these  limiting  values 


§73a.]  FORMULAS   FOR   T-BEAMS.  87 

as  approximations  for  the  true  ones,  we  have  as  substitutes 
for  (7)  and  (8) 


•W     j 
-JO  J 


(7)' 


a   T     M       f    T     f    20  «v 

0=T=>    f'='   /c=         •    •    • 


The  errors  involved  in  these  approximations  are  on  the  side  of 
safety,  for  (7)'  gives  values  smaller  than  (7),  and  (8)'  larger 
ones  than  (8).  Satisfactory  approximate  results  may  also  be 
reached  by  assuming  a  fixed  value  of  Id  for  the  arm  of  the 
resisting  couple  jd. 

7$a.  Formulas  Taking  into  Account  the  Compression  in  the 
Web.—  When  the  web  is  very  large  compared  to  the  flange  it 
may  be  desirable  to  use  more  exact  formulas  than  those  already 
given.  In  this  case  the  formulas  for  the  position  of  neutral 
axis,  arm  of  resisting  couple,  and  moment  of  resistance  be- 
come as  follows  : 


2ndA^-b^     /nA  +  (b-b')t\*    nA  +  (b-b')t 
x~      ~~        +  '       ~~ 


V 


bt(2kd-t)+b'(kd-t)* 

jd=d-z-,     .......     (11) 


fc\  1 

Mc  =  Md  I  (2M  ~Vbt+  (  W  ~  Wb'  i 
Equations  (12)  also  give  /s  and  fc  for  given  values  of  M. 


88  GENERAL    THEORY.  [Cn.  III. 

74.  Problems  of  Design — Either  Case  I  or  II. — In  practice, 
various  forms  of  problems  will  arise:  (a)  The  dimensions  may 
be  given,  to  find  the  safe  resisting  moment  of  the  beam  or 
the  stresses  in  the  steel  and  concrete  under  a  given  load;  (b) 
the  dimensions  of  the  flange  may  be  given,  together  with 
the  loading  and  specified  working  stresses,  to  determine  suit- 
able web  dimensions  and  steel  area;  (c)  the  loading  and  work- 
ing stresses  may  be  given,  to  determine  suitable  proportions 
for  the  entire  beam. 

(a)  Where  all  the  dimensions  are  given,  the  value  of  k 
and  j  are  found  from  eqs.  (3)  and  (6),  or  from  Plate  VI,   and 
thence  the  values  of  the  moment  of  resistance  from  eqs.  (7), 
or  the  fibre  stresses  from  (7)  or  (8).     If  the  value  of  k  is  found 
to  be  less  than  t/d  then  the  problem  falls  under  Case  I  and 
the  formulas  for  rectangular  beams  apply,  or  the  approximate 
formulas  of  Art.  72  may  be  used. 

(b)  Generally  the  flange  has  been  predetermined  as  it  is 
usually  formed  by  a  portion  of  the  floor  slab  which  is  already 
designed.  '  A  suitable  web  must  then  be  determined,  together 
with  the  necessary  amount  of  steel;  and  finally  the  fibre  stress 
in  the  concrete  must  be  calculated  to  ascertain  if  it  is  within 
the  specified   working  limit.     The  depth  and  width  of  web 
are  selected  with  reference  to  shearing  strength,  space  for  the 
necessary  rods  and  other  considerations,  as  fully  explained  in 
subsequent    articles.     The    depth    having    been    selected,    the 
value  of  j  is  estimated  and  the  amount  of  steel,  A,  approxi- 
mately determined   by  eq.    (7).     The  amount  of  steel  being 
known,  the  value  of  j  can  be  determined  by  eq.  (6)  and  then, 
if  necessary,  the  value  of  A  corrected  by  eq.  (7).     The  value 
of  k  should  also  be  found  from  eq.  (2)  in  order  to  ascertain  if 
the  beam  falls  under  Case  I  or  II.     The  stress  in  the  concrete 
is  then  found  from  eq.  (8). 

In  estimating  the  value  of  j  use  Plate  VI,  or  a  value  of  | 
may  be  assumed,  as  for  rectangular  beams. 

(c)  When  all  parts  of  the  beam  are  to  be  selected  on  the 


§74.]  FORMULAS    FOR   T-BEAMS.  89 

basis  of  given  working  stresses  it  is  convenient  to  first  select 
suitable  proportions  for  the  web,  as  in  Case  (6).  A  flange 
thickness  is  then  assumed  such  as  to  give  satisfactory  pro- 
portions between  t  and  d.  The  value  of  t/d  is  then  known 
and  k  and  j  can  be  determined  from  (1),  (4),  and  (5).  The 
area  of  steel  and  the  breadth  of  flange  is  then  found  from 
eq.  (7).  The  smaller  the  value  of  I  the  smaller  will  be  the 
flange  area  required,  but  too  slender  proportions  are  to  be 
avoided,  as  explained  in  Chapter  V 

Examples— (I)  A  T-beam  has  the  following  dimensions:  6=48  in., 
1=6  in.,  d  =  22  in.,  and  b'  =  10  in.;  the  steel  consists  of  six  f-in.  rods. 
If  the  working  strengths  of  steel  and  concrete  are  15,000  and  600  Ibs/in* 
respectively,  and  n=15,  what  is  the  safe  resisting  moment  of  the  beam? 

Solution.  The  area  of  the  steel  is  2.65  in',  and  p  =  2.65 /  (48X22)  = 
0.0025.  Supposing  this  beam  to  fall  under  Case  I,  we  find  k  from  Fig.  17 
(or  eq.  (1),  Art.  55)  to  be  about  0.24,  hence  kd  =  5.3  in.,  and  the  neutral 
axis  is  in  the  flange,  that  is,  the  case  was  correctly  guessed.  Now 
y=l  -p  =  0.92;  hence  (see  eqs.  (3)  and  (4),  Art.  56) 

/, 

Ms=  (15,000X2.65) (0.92X22)  =806,000  in-lbs., 

and  Mc  =  300(5.3X48) (0.92X22)  =  1,545,000  in-lbs., 

The  safe  resisting  moment  hence  depends  on  the  steel,  as  it  usually 
does  in  T-beams.  The  approximate  formula  gives  Ms  =  795,600  in-lbs. 

(2)  Change  t  of  the  preceding  example  to  4  in.  and  find  the  safe 
resisting  moment. 

Solution.  Evidently  this  beam  now  falls  under  Case  II.  Equation 
(2)  gives  k  =  0.247  and  (4)  2  =  1.61  in.  From  (7) 

'   y      v        '-)•'.:•* 

Ms=  15,000X2.65(22 - 1.61)  =950,000  in-lbs., 

4 

and  Mc=  (600/5.44)3.44(48X4) (22  -  1.61)  =  1,485,000. 

The  values  of  k  and  ;  may  also  be  found  from  Plate  VI. 

The  approximate  formulas  (7)'  give  Afs  =  716,000  and  Afc  =  1,152,000 
in-lbs. 

(3)  Suppose  that  the  diameter  of  the  rods  in  example  (1)  is  1  in,, 


90  GENERAL    THEORY.  [Cn.  III. 

and  that  the  beam  is  subjected  to  a  bending  moment  of  1,250,000  in-lbs. 
Compute  the  working  stresses  in  the  steel  and  concrete. 

Solution.     Equation    (2)    gives   k  =  0.306   and   fcd  =  6.73,   hence   the 
beam  falls  under  Case  II.     Equation  (4)  gives  2  =  2.22  in.,  and  (8) 


,     13,400    0.306 
and  /'—     "  =  395  lbfl/m  ' 


The  approximate  formulas  (8)'  give /s  =  13,960  and /c  =  457  lbs/in3. 

(4)  The  flange  of  a  T-beam  is  24  in.  wide  and  4  in.  thick.     The 
beam  is  to  sustain  a  bending  moment  of  480,000  in-lbs.,  the  working 
strengths  of  steel  and  concrete  being  respectively  15,000  and  500  lbs/in2. 
What  depth  of  beam  and  amount  of  steel  will  answer? 

Solution.  We  will  try  d  =  lS  in.  Assume  jd  =  !Q  in.  Then  eq.  (7) 
gives  A  =  2  in2,  and  hence  p  =  2/  (24  X 18)  =  0.00462.  Then  (6)  gives 
/=0.91  and  the  corrected  value  of  A  is  1.95  in2.  Equation  (3)  gives 
k  =  0.325,  and  shows  that  the  beam  falls  under  Case  II  as  assumed. 
The  stress  in  the  concrete  is  found  from  (8)  to  be  (15,000/15)  X  0.325/0.675 
=  480  lbs/in'',  which  is  permissible. 

(5)  Let  it  be  required  to  design  a  T-beam  to  sustain  a  bending 
moment  of  480,000  in-lbs.,  the  working  stresses  to  be  15,000  and  600 
lbs/in5,  respectively. 

Solution.  The  same  depth  of  web  and  thickness  of  flange  will  be 
assumed  as  in  (4),  as  these  proportions  are  reasonable.  The  amount 
of  steel  and  width  of  flange  are  to  be  determined.  From  (1)  we  find 
fc=  1^(15,000/9000  +  1)  =0.375,  and  from  (4)  2=1.72  and  ;d=18.0- 
1.72  =  16.28  in.  Then  from  (7), 

4  =  1.97  in2    and    b  =  17.5  in. 

74a.  Diagrams  of  M/bd2  for  Use  in  Designing.— Ey  reason 
of  the  additional  variable  (the  flange  thickness)  involved  in 
formulas  relating  to  T-beams  as  compared  to  rectangular 
beams,  it  is  not  possible  to  arrange  so  simple  a  graphical 
solution  of  the  resisting  moment  as  is  done  by  means  of  Plates 
I-IV,  Chapter  VI.  Assuming,  however,  a  single  value  of  n 


§74a.]  FORMULAS    FOR   T-BEAMS.  91 

the  values  of  the  resisting  moment  or  coefficient  of  resistance 
may  readily  be  represented  graphically.  It  will  be  convenient 
to  consider  as  variables  the  values  of  fa,  fe,  and  the  ratio  t/d, 
or  thickness  of  slab  to  the  effective  depth  of  the  beam.  From 
eq.  (7)  we  may  write 


»-'•('-  a)  r* 


In  this  equation  k  and  /  are  functions  of  fa  and  fc,  as  appears 
from  eqs.  (1),  (4),  and  (5).  Plates  VII-XI  are  plotted  from 
this  equation,  assuming  n  =  15  in  all  cases.  Each  plate  contains 
values  of  M  ,/bd2  for  a  certain  value  of/,  and  for  various  values 
of  fe  and  t/d.  On  the  same  diagram  are  also  given  values  of 
k  and  values  of  j.  The  former  are  given  by  the  dotted  curve 
in  the  right-hand  part  of  the  diagram,  and  the  value  corre- 
sponding to  a  given  value  of  fc  is  to  be  read  off  on  the  axis 
for  t/d.  This  value  of  k  is,  in  fact,  the  value  of  t/d  which 
brings  the  neutral  axis  just  to  the  lower  surface  of  the  flange. 
These  diagrams  are  particularly  useful  in  solving  problems 
under  cases  (6)  and  (c),  Art.  74;  they  are  not  adapted  to 
case  (a).  Plate  VI  gives  all  the  information  needed  for  this 
case. 

Examples.—  Examples    (4)    and    (5)    given  in   the   previous  article 
will  now  be  solved  by  means  of  these  diagrams. 

(4)  Use  Plate  IX.     The  value  of  M/6d2  =  480,000/(24X182)=61.6. 
For  this  value  of  M/bcF  and  for  t/d=4./lS,  we  find  from   the  diagram 
/c  =  about  470  lbs/in?  and  /  =  .9L    Then,  as  before,  A  =480,000/(15,OOOX 
.91X18)  =  1.95  in2 

(5)  Use  Plate  IX.     For/c  =  600  and  t/d  =  4/lS,  we  find  M/M'  =  85, 
whence  b  -480,0007  (85  X182)  =  17.4  in.;  also,  /  =  .90  and  A  =  1.97  in2. 

(6)  Using  the  same  depth  of  beam  as  in  Ex.  (5)  what  will  be  the  effect 
on  the  amount  of  concrete  and  steel  of  changing  the  flange  thickness  to 
3  inches  and  to  5  inches  ?   From  Plate  IX  we  find,  for  t/d  =  3/18,  M/bd2  = 
74,  whence  6  =  20.0  in.,  /  =  .92  and  A  =  1.93  in2.    For  </d  =  5/18,  M/bd*=* 
93,  6  =  16.0  in..  /  =  .91  and  A  ==  1.95  in2.    The  volumes  of  the  concrete  in 
the  flanges  are,  for  the  flange  thicknesses  of  3,  4  and  5  in.,  respectively, 
60,  69.6,  and  80  sq.  in.    The  steel  areas  vary  but  little. 


92  GENERAL  THEORY.  [CiJlII.. 


V 


75.  T-beams  Double-reinforced.— T-beams  are  often !( 
tinuous  over  their  supports ;  at  such  places  the  bending 

is  negative,  and  the  flange  is  under  tension  and  the  lower  part 
of  the  web  under  compression.  Not  only  is  tensile  steel  pro- 
vided, but  some  steel  is  always  left  in  the  web  (see  Chap.  VII) ; 
that  is,  the  beam  is  reinforced  in  compression,  and  is  said  to  be 
double-reinforced.  For  a  discussion  of  double-reinforcement 
see  the  following  articles — particularly  Art.  79 — in  which  there 
is  explained  a  simple  method  for  determining  the  effect  of  the 
compressive  steel  on  the  stress  in  the  tensile  steel  and  the  com- 
pression in  the  concrete. 

76.  Beams  Reinforced  for  Compression. — The  compression 
in  the  concrete  is  assumed  to  follow  the  linear  law  and  the 
tension  in  .it  is  neglected ;   the  formulas  then  apply  to  working 
conditions  only.    In  addition  to  the  notation  already  adopted 
(see  page  58),  let 

A'  denote  the  cross-sectional  area  of  the  compressive  re- 
inforcement; 

pr  denote  the  steel  ratio  for  the  compressive  reinforcement, 
that  is  A'/bd] 

}g  denote  the  unit  stress  in  the  compressive  reinforcement; 

Cf  denote  the  whole  stress  in  the  compressive  reinforcement; 

d'  denote  the  distance  from  the  compressive  face  of  the 
beam  to  the  plane  of  the  compressive  reinforcement ; 

x  denote  the  distance  from  the  compressive  face  to  the 
resultant  compression,  C+  C',  on  the  section  of  the  beam. 

77.  Neutral  Axis  and  Arm  of  Resisting  Couple, — From  the 
x  Stress  diagram  (Fig.  28)  it  appears  that  f8/nfc=(d—kd)/kd}  or 


/«-     >-JT*». 

Similarly,  ////i/c  =  (kd-d')/l:d,  or 


k-d'/df 
fs'=n — —fc (2) 


§77.] 


BEAMS  DOUBLE-REINFORCED. 


93 


For  simple  flexure,  the  whole  tension  T  and  whole  compression 
C+C'  are  equal;  hence 

'  .......     (a) 


Inserting  the  values  of  }s  and  //  from  (1)  and  (2)  in  (a)  gives 
an  equation  which  may  be  written  thus: 


'd'/d),  ....     (3) 


FIG.  28. 

and  from  this  the  neutral  axis  of  a  given  section  can  be  located. 
The  low^er  group  of  curves  in  Fig.  29  gives  values  of  k  for  several 
values  of  p  and  all  values  of  p'  up  to  2%;  n  is  taken  at  15 
and  d'/d  as  1/10.  Thus  for  p  =  2%  and  p'  =  1.5%,  k  =  0.434. 
The  arm  of  the  resisting  couple  is  the  distance  between  T 
(see  Fig.  28)  and  the  resultant  of  the  compressions  C  and  C'. 
It  follows  from  the  principle  of  moments  and  the  law  of  dis- 
tribution of  stress  respectively  that 


l  +  C'/C 


d} 


.     C'    2p'n(k-d'/d) 

and     -= 


from  which  z  can  be  computed  for  any  given  section, 
the   arm   jd  =d  —z  or 


Finally 


(4) 


The  upper  group  of  curves  in  Fig.  29  gives  values  of  /  for  sev- 
eral values  of  p  and  all  values  of  p'  up  to  2%;  n  is  taken  at 
15  and  d'/d  at  1/10.  Thus  for  p=2%  and  ?/  =  1.5%,  /=0.875. 


94 


GENERAL  THEORY, 


[Cn.  III. 


All  Values  of  j  and  k  are  based  on 
W-15  and     d^rd-  VlO 


Percentag  5  of  Compressiye  Steel 


PIG.  29. 


§78.]  BEAMS  DOUBLE-REINFORCED.  95 

78.  Resisting  Moment  and  Working  Stresses.  —  If  the  tensile 
reinforcement  is  low,  the  resisting  moment  depends  upon  it, 
and  is  given  by 

......     (5) 


If  the  compressive  reinforcement  is  low,  the  resisting  moment 
depends  upon  !t  and  the  concrete,  and  is  given  by 

c  =  i/c&(l  -  ±k)bd2  +  fs'p'bd(d-d')  ; 


c 

but  //  bears  a  certain  relation  to  fc  (see  eq.  2),  which  inserted 
in  the  preceding  equation  gives  finally 

Mc  =  [k(±-±k)  +  np'(k-df/d)(l-d'/d)/k]fcbd2.  .     .     (6) 

The  unit  fibre  stress  in  the  tensile   steel  produced  by  any 
bending  moment  M  can  be  computed  from 


_ 
s~    A    ~pjbd2' 

and  those  in  the  concrete  and  compressive  steel  from  Js  and 
equations  (1)  and  (2)  respectively. 

Fig.  29  shows  that  the  neutral  axis  is  nearer  the  compressive 
steel  (k  <  0.55)  unless  the  percentage  of  tensile  reinforcement 
is  quite  high  and  the  compressive  low;  thus  for  p  =  3%,  the 
neutral  axis  is  nearer  the  compressive  steel  unless  p'  is  less 
than  3/4%,  and  when  p  =  2%,  it  is  nearer  for  all  values  of 
pf  '.  Now  since  the  unit  stresses  in  the  tensile  and  compressive 
steels  are  as  the  distances  of  the  steels  from  the  neutral  axis, 
it  follows  that  the  unit  stress  in  the  compressive  steel  is  gen- 
erally less  than  that  in  the  tensile,  that  is  /,'</,. 

For  approximate  computations  one  might  use  the  average 
values  /  =  0.85  and  k  =  OA5  in  equations  (5),  (6),  and  (7); 
then  they  would  become  respectively  (n=15) 

(5)' 

.....     (6)' 
(7)' 


96  GENERAL  THEORY.  [Cn.  III. 

79.  Determination  of  Amount  of  Compressive  Reinforce- 
ment. —  This  problem  presents  itself  as  follows:  From  the  cir- 
cumstances of  the  case,  the  beam  needs  so  much  tensile  steel 
that  the  compressive  concrete,  if  unreinforced,  would  be  stressed 
too  high,  and  it  is  necessary  to  employ  compressive  reinforce- 
ment to  reduce  the  stress  in  the  concrete;  the  percentage  of 
reinforcement  necessary  to  lower  the  stress  a  certain  amount 
is  desired. 

An  explicit  formula  for  this  percentage  is  too  cumbersome 
for  practical  use,  but  a  diagram  (Plate  VI,  page  280)  can  be 
constructed  from  which  the  desired  quantity  can  be  easily 
determined.  The  construction  of  such  a  diagram  will  now  be 
explained. 

Let  fa  and  fc  denote  the  unit  stress  in  the  tensile  steel  and 
the  concrete  respectively,  kd  the  depth  of  the  neutral  axis, 
and  jd  the  arm  of  the  resisting  couple,  CT,  when  there  is  no 
compressive  reinforcement  (see  Fig.  16);  also  let  /,',  /«/,  k'dt 
and  j'd  denote  the  same  quantities  when  there  is  compressive 
reinforcement.  Then 

fskd  Mk 

'c~ 


n(d-kd)~jdAn(l-k)' 

fs'k'd  Mk' 

n(d-kfd)    jfdAn(l-k'Y 


From  these  the  relative  reduction  in  fc  due  to  the  addition  of 
compressive  steel  is  found  to  be 


/„          f  k  i-kr 


/« 


Since  /  and  k  depend  on  p,  and  /'  and  kf  on  pf,  the  equation 
furnishes  the  relation  between  relative  reduction  in  concrete 
stress  and  the  percentages  of  steel.  The  relative  reduction 
(fc—fc)/fc  depends  largely  on  the  percentage  of  compressive 
steel  and  for  a  given  value  of  this  percentage  the  reduction  is 
practically  the  same  for  all  ordinary  percentages  of  tensile 
steel  (from  J  to  3%).  Plate  VI,  page  280,  gives  values  of 


§79.]  BEAMS  DOUBLE-REINFORCED.  97 

this  reduction  for  different  values  of  compressive  steel  from 
0  to  2%.  As  heretofore,  values  n  =  15  and  d'/d  =  l/10  were 
used. 

Addition  of  compressive  steel  reduces  the  stress  in  the  ten- 
sile steel.    The  relative  amount  of  this  reduction  is  given  by 


The  group  of  curves  (Plate  XII)  gives  this  reduction  in  per 
cent  (right-hand  margin)  for  different  percentages  of  tensile  and 
compressive  steels  as  noted.  (For  illustration  of  the  use  of 
this  diagram,  see  example  (3)  following.) 

Examples.  —  (1)  A  beam  of  which  6  =  12  in.,  d  =  18  in.,  and  d'/d=Vio 
has  2£%  of  tensile  steel  and  1%  of  compressive.  If  the  working  strengths 
of  steel  and  concrete  are  15,000  and  600  lbs/in2  respectively,  what  is 
the  safe  resisting  moment  of  the  beam? 

Solution.    From  Fig.  29,  fc  =  0.5  and  /=0.85;   therefore 

M,  =  15,000  X  0.025  X0.85  X  12  X  182  =  1,238,000  in-lbs., 
and 
Mc=  (0.5  X0.417  +  15X0.01X0.4X0.9/0.5)600X12X182  =  736,000  in-lbs., 

which  is  the  safe  resisting  moment. 

(2)  Suppose  that  the  beam  of  the  preceding  example  were  subjected 
to  a  bending  moment  of  1,000,000  in-lbs.    What  are  the  working  stresses 
/c,  /a,  and  /.'? 

Solution.  As  in  example  (1),  A;  =  0.5  and  /=0.85;  therefore  (see 
eq.  7)  /«=1;000,000/0.025X0.85X12X182  =  12,100  lbs/in2.  From  equa- 
tion (1),  /c  =  (12,100X0.5)  -(15X0.5)  =  810  lbs/in2,  and  from  equation 
(2),  //  =  15  (0.4/0.5)810  =  9720  lbs/in2. 

(3)  In  a  certain  design  of  a  beam  it  is  necessary  to  use  2.5%  of 
tensile  steel  and  this  would  result   in  a  stress  of  1200  lbs/in2  in  the 
concrete;   it  is  necessary  to  reduce  this  to  900  by  adding  compressive 
steel.     How  much  additional  steel  is  required? 

Solution.  (See  Plate  XII.)  The  desired  reduction  of  the  compressive 
stress  is  25%.  We  find  this  value  at  the  left  side  of  the  diagram,  then 
trace  horizontally  to  the  concrete  curve,  and  then  down  to  the  lower 
margin,  reading  there  0.9%,  the  required  quantity.  From  the  last  point 
we  trace  up  to  the  2.5%  steel  curve  and  then  to  the  right  margin,  where 
we  note  about  4.5%  reduction  in  tensile  steel  stress  due  to  0.9%  com- 
pressive steel. 


98 


GENERAL  THEORY. 


[Cn.  in. 


80.  Flexure  and  Direct  Stress. — When  the  resultant,  R,  of 
the  external  forces  acting  on  one  side  of  a  section  of  a  beam  is 
not  parallel  to  the  section,  then,  in  general,  there  exist  both 
direct  and  flexural  stresses  at  the  section.  The  exception 
obtains  when  the  resultant  passes  through  the  centroid  of  the 
section  (transformed,  as  explained  below,  if  the  section  is  rein- 
forced unsymmetrically) ;  in  this  exceptional  case  the  fibre 
stress  is  wholly  direct. 

In  concrete  work,  the  direct  stress  is  always  compressive. 
Combination  of  direct  compressive  and  flexural  stress  gives 
resultant  fibre  stress  which  is  either  (1)  all  compression  or  (2) 
part  compression  and  part  tension;  these  cases  are  discussed 
separately  below.  Whether  a  given  R  will  produce  fibre  stress 
falling  under  case  (1)  or  (2)  depends  on  the  eccentricity  *  of  R, 
the  relative  amounts  of  steel  and  concrete  at  the  section  and 
on  n.  If  the  reinforcement  is  symmetrical,  steel  imbedded  a 
depth  equal  to  1/10  the  whole  depth  of  beam,  and  n  is  15,  then 
for  eccentricities  lower  than  those  given  in  the  table,  case  (1) 
obtains,  and  for  higher,  case  (2). 


p= 

0% 

107 
$70 

1% 

u% 

2% 

e/h= 

i 

0.187 

0.202 

0.214 

0.224 

In  addition  to  notations  already  adopted,  the  following  will  be 
used  (see  Fig.  30) : 

R  denotes  the  resultant  of  all  the  external  forces  acting  on 
a  beam  on  either  side  of  the  section  under  considera- 
tion; 

6  denotes  the  eccentric  distance  of  R',   that  is,  the  distance 
from  the  point  where  R  cuts  the  section  to  the  middle 
of  the  section; 
N  denotes  the  component  of  R  normal  to  the  section; 


*  By  the  eccentricity  of  R  is  meant  the  ratio  of  the  distance  between 
the  centre  of  the  section  and  the  point  where  R  pierces  the  section  to  the 
whole  height  of  the  section. 


§81.]  FLEXURE  AND  DIRECT  STRESS.  99 

M  denotes  bending  moment  at  the  section;  it  equals  Ne  or 
the  sum  of  the  moments  of  all  the  external  forces  about 
the  horizontal  line  through  the  middle  of  the  section, 
but  when  the  transformed  section  is  used,  the  moment 
axis  must  be  taken  through  its  centroid; 
A'  denotes  the  area  of  the  steel  nearer  the  face  of  the  con- 
crete most  highly  stressed; 
d'  denotes  the  distance  from  that  face  to  the  plane  of  this 

steel ; 

A  denotes  the  area  of  the  steel  at  the  other  face; 
d  denotes  the  distance  from  the  former  face  to  the  plane  of 

this  steel; 

h  denotes  the  whole  height  of  the  section; 
pf  denotes  the  steel  ratio  A'/bh; 
p  denotes  the  steel  ratio  A/bh; 
u  denotes  the  distance  from  the  face  most  highly  stressed 

to  the  centroid  of  the  transformed  section; 
At  denotes  the  area  of  the  transformed  section; 
It  denotes  the  moment  of  inertia  of  the  transformed  section 

with  respect  to  its  horizontal  centroidal  axis; 
Ic  denotes  the  moment  of  inertia  of  the  section  bh  with 

respect  to  that  axis;  and 
I8  the  moment  of  inertia  of  the  sections  of  the  steel  with 

respect  to  the  same  axis. 

81.  Transformed  Section. — By  the  transformed  section  of  a 
reinforced  concrete  beam  is  meant  the  actual  section  with  the 
areas  of  the  reinforcement  replaced  by  concrete  n-fold  and  in 
the  planes  of  the  reinforcement.  Thus  if  Fig.  30 a  represents 
an  actual  section,  306  represents  the  section  transformed,  the 
areas  of  the  upper  and  lower  wings  of  the  latter  section  being 
respectively  n  times  the  areas  of  the  upper  and  lower  rein- 
forcements. 

(A  prism  of  steel  of  a  given  area  and  one  of  concrete  of  n 
times  the  area  are  equally  rigid  as  regards  simple  tension  or 
compression;  hence  a  reinforced-concrete  beam  and  a  plain 
concrete  beam  whose  section  is  that  of  the  first  transformed  are 


100 


GENERAL  THEORY. 


[Cn.  III. 


equally  stiff  in  so  far  as  stiffness  depends  upon  fibre  stress,  and 
in  certain  cases,  as  stated  later,  the  fibre  stress  in  the  reinforced 
beam  can  be  computed  from  those  in  the  plain  concrete  beam. 
In  those  cases,  the  actual  section  and  the  transformed  section 
are  equivalent,  ideally  at  least.  Actually,  the  two  beams  are 


(a) 


FIG.  30. 


not  equally  strong  because  of  dangerous  stresses  in  the  wings 
of  the  transformed  section.) 

Referring  to  Fig.  30  it  will  readily  be  seen  that 


At-bh+n(A+A'),    /,=/c+n/t, 


M 


h/2+npd+np'd' 


1  +np  +np' 

Ic=$b[u3  +  (h-u)*]    and    I9=A(d-u 
If  the  reinforcement  is  symmetrical,  then  u  =  h/2  and 


•     (2) 
'•     (3) 


and 


(3)' 


82.  Case  I.  The  Fibre  Stress  is  Wholly  Compressive. — (a) 
The  unit  fibre  stress  in  the  concrete  can  be  computed  just  as 
though  the  beam  were  homogeneous,  but  the  transformed 
section  must  be  used  in  the  computations  if  the  beam  is  rein- 
forced. The  unit  stresses  in  the  steel  will  be  n  times  those  in 
the  concrete  in  the  planes  of  the  reinforcements  respectively. 
Thus  the  unit  direct  stress  in  the  concrete  is  N/At'}  the  unit 
flexural  stress  in  the  concrete  highest  stressed  is  Mu/It;  that 
in  the  concrete  adjoining  the  reinforcement  highest  stressed 


I  82.] 


FLEXURE  AND  DIRECT  STRESS. 


101 


is  M(u— d')/lt]  and  that  in  the  concrete  adjoining  the  other 
reinforcement  is  M(d—u)/It.    The  combined  unit  stresses  are: 


N     Mu 


N_    nM(u-d') 
N     nM(d-u) 


(4) 
(5) 
(6) 


These  equations — and  the  stress  diagram,  Fig.  31 — show  that 
/,  is  always  less  than  //,  and  //  is  always  less  than  n/c;  hence 
the  unit  stresses  in  both  steel  reinforcements  will  always  be 
safe  if  fe  is  a  safe  value. 


ld'ti—J—1         T 

-^::  I 


Fio.  31. 


(6)  The  method  employed  for  simple  flexure,  suitably  modi- 
fied, leads  to  formulas  not  involving  the  transformed  section, 
as  will  now  be  explained. 

From  the  stress  diagram  (Fig.  32)  it  will  be  seen  that 


and 


f.'  =  nfc(l-d'/kh), 
/.  -n/fl(l-d/to), 
//  =/.d-  1/*).  . 


(7) 
(8) 
(9) 


102  GENERAL  THEORY.  [Cn.  III. 

From  the  condition  that  the  resultant  fibre  stress  equals  N, 


and  from  the  condition  that  the   moment  of  the-  total   fibre 
stress  about  the  centroidal  axis  equals  M, 


From  these  equations  it  is  possible  to  compute  the  unit  fibre 
stresses  fc,  fs,  and  /,'  in  a  given  case. 

When  the  reinforcement  is  symmetrical  the  equations  simplify 
greatly,  and  they  lead  to  the  following  formula  : 

I2k(l  +2np)e/h  =  l  +24npa2/h2  +6(1  +2np)e/h;    .     (10) 
they  also  give  the  following  formula  for  fc  or  M  : 


When  d'/h  =  l/W,  and  n  =  15,  Fig.  33  gives  values  of  1/k 
for  different  values  of  eccentricity  and  percentage  of  steel; 
thus  for  e/h  =  0.1  and  p  =  l  .5%,  l/k  =  0.635,  hence  k  =  1.57. 

83.  Case  II.  There  is  Some  Tension  at  the  Section.  —  (a)  If 
the  tension  in  the  concrete  is  so  small  as  to  be  permissible,  and 
this  tension  is  taken  account  of  in  the  computations,  then  the 
unit  fibre  stresses  in  the  concrete  and  steel,  if  reinforcement 
is  present,  may  be  computed  by  the  method  explained  under 
Case  I.* 

The  combined  unit  stress  in  the  remote  tensile  fibre  is  given 

by 


_ 

It          A,' 


*  It  is    assumed  that  the  linear  law  of  variations  of  the  unit  flexural 
stresses  holds  for  the  tension  as  well  as  compression. 


§83.] 


FLEXURE  AND  DIRECT  STRESS. 


103 


0.05 


0.10 


0.15 


X 


X 


/ 


/////. 


IL 


lin 


III 


"a  jo. 


ILL 


All  Values  of  k  are 
based  on  n=l5  and 


kh 


I 


III 


II 


1 


0.05 


0.10 
Eccentricity  e/h, 

FIG.  33. 


O.'IS 


104 


GENERAL  THEORY. 


[CH.  IIT. 


and  fs  as  given  by  (5)  is  compressive  or  tensile  according  as  its 
value  is  positive  or  negative. 

(6)  If  the  tensile  stresses  are  so  high  that  it  is  advisable  to 
neglect  the  tension  in  the  concrete,  then  a  method  similar  to 
that  used  heretofore  in  simple  flexure  is  simplest.  The  trans- 
formed section  is  not  used.  0  (Fig.  34)  denotes  a  horizontal 


FIG.  34. 

axis  at  mid-depth  of  the  beam,  M  the  moment  sum  of  all  the 
external  forces  on  one  side  of  the  section  with  respect  to  that 
axis,  and  N,  as  before,  the  algebraic  sum  of  the  components 
of  those  forces  perpendicular  to  the  section.  From  the  stress 
diagram,  it  follows  that 


and 


(13) 


Since  the  resultant  fibre  stress  equals  N, 

.      $fcbkh+fs'A'-f8A=N, 

and  since  the  moment  of  the  fibre  stress  about  the  horizontal 
axis  through  0  equals  M, 


From  these  four  equations  k,  }c,  fs,  and  //  can  be  determined 
for  a  given  section,  reinforcement,  M,  and  N. 


§84.]  FLEXURE  AND  DIRECT  STRESS.  105 

//  the  reinforcement    is   symmetrical,    then    the    equations 
simplify.     The  value  of  k  is  given  by 

.     .     (15) 

The  greatest  unit  compressive  fibre  stress  in  the  concrete  is 
given  by 


and  the  unit  stresses  in  the  steel  are  given  by  (7)  and  (8). 
From  (7),  or  the  stress  diagram,  it  is  plain  that  /,'  is  less  than 
nfc  even  for  unsymmetrical  reinforcements. 

When  df/h  =  l/10  and  ft  =  15,  Fig.  35  gives  values  of  k  for 
different  values  of  eccentricity  and  percentage  of  steel;  thus 
for  e/h  =  l,  andp=0.8%,  £  =  0.42. 

84.  Diagrams.  —  To  facilitate  the  application  of  equation  (11) 
(Case  I),  and  equation  (16)  (Case  II),  Plates  XIII  and  XIV, 
pages  287  and  288,  have  been  constructed. 

In  the  first  diagram,  values  of  the  eccentricity,  e/h,  are 
given  at  the  upper  and  lower  margins;  the  ordinates  from 
the  lower  margin  to  any  curve  are  values  of  (l  +  24:npa2/h2)/l2k 
(see  equation  11),  and  hence  of  M/bh2}c  ,for  the  value  p  marked 
on  that  curve.  Thus  when  e/h  =  0.1  and  p  =  1%,  M/bh2fc  =  0.087. 

The  dotted  portions  of  the  curves  correspond  to  eccen- 
tricities which  involve  small  tensile  stress  in  the  concrete  and 
belong  strictly  to  Case  II.  The  values  of  the  unit  tensile  stress 
//  can  be  calculated  from  equation  (12)  or  from 


fe       kh 


l/k  being  obtained  from  equation  (10),  or  from  an  extension  of 
the  appropriate  curve  in  Fig.  33. 

In  the  second  diagram,  also,  values  of  the  eccentricity  e/h 
are  given  at  the  upper  and  lower  margins;  the  ordinates 
from  the  lower  margin  to  any  solid  curve  are  values  of 


106 


GENERAL  THEORY. 


[Cn.  III. 


Eccentricity, 
0.5 


,  for  Upper  Group  of  Curves. 
10  _        15 


All  Values  of  A:  are 
on  n=  15  and  d'-yfr- 


0.2  0.3 

Eccentricity,  e/h,  for  Lower  Group  of  Curves. 

FIG.  35. 


§85.]  FLEXURE  AND  DIRECT  STRESS.  107 

-&k(3-2k)+2pna2/kh2  (see  equation  16),  and  hence  of 
M/bh2fc,  for  the  value  of  p  marked  on  that  curve.  Thus 
when  e/h  =  l  and  p  =  l%,  M /bh*fc  =  0.187. 

The  dotted  curves  in  the  second  diagram  enable  one  to 
estimate  the  ratio  of  the  unit  stress  in  the  tensile  steel  to  that 
in  the  concrete,  /«//c,  for  most  eccentricities  and  percentages 
of  steel.  Thus  when  e/h  =  l  and  p=0.5%,  we  find  e/h  =  l 
at  the  top  or  bottom  and  then  trace  vertically  to  the  0.5% 
curve  and  note  the  point  of  intersection.  This  point  falls 
between  the  curves  /,//c=20  and  25,  and  the  ratio  is  about  21. 
For  values  of  e/h  and  p,  which  bring  the  "point "  to  the  left 
of  the  line  /,//c=15,  /,  will  be  less  than  15/c,  and  hence  less 
than  the  working  strength  of  steel  for  all  ordinary  allowable 
values  of  fc.  No  similar  curves  for  f8/fc  appear  on  the  first 
diagram  because  that  ratio  is  always  less  than  15,  and  hence 
the  unit  stresses  in  the  steel  (both  upper  and  lower)  are  within 
safe  values  for  Case  I,  if  fc  is  safe. 

85.  Examples. — It  is  supposed  hi  these  that  the  steel  is 
imbedded  a  depth  of  one-tenth  the  total  height  of  the  beam, 
and  that  n  =  15,  so  that  the  diagrams  on  pages  287  and  288  apply. 

(1)  A  beam  is  12  in.  wide,  30  in.  high,  and  contains  1%  of  steel 
above  and  an  equal  percentage  below.    At  a  particular  section,  the 
resultant  R  is  80,000  Ibs.,  its  inclination  to  the  axis  of  the  beam  is  5°, 
and  its  eccentric  distance  is  4.5  in.    Compute  the  unit  fibre  stresses 
in  the  concrete  and  steel  (/c,  /«,  and  /«'). 

Solution.  The  eccentricity  is  e/h = 0.15,  and  M = 80,000  cos  5C  X  4.5  = 
358,650  in-lbs.  The  beam  falls  under  Case  I  because  this  eccentricity 
gives  a  "point"  on  the  \%  curve  of  page  287,  but  not  on  that  of  page  288. 
Tracing  horizontally  from  the  point  we  read  M/bh%= 0.112;  hence 

358,650 

/C=24X30'X0.112°297  Ib8/in  ' 

The  unit  stresses  hi  the  steel  are  less  than  15/c=4500  lbs/in2.  Their 
exact  values  can  be  computed  from  equations  (7)  and  (8);  the  value  of 
A;  for  use  in  them  can  be  easiest  obtained  from  the  diagram  on  page  103. 

(2)  Change  the  eccentricity  of  the  preceding  example  to  15  in.  and 
solve. 

Solution.  The  eccentricity  is  e/h = 0.5,  and  M = 80,000  cos  5°  X 15  = 
1,195,500  in-lbs.  The  beam  falls  under  Case  II  (see  page  288),  and  for 


108  GENERAL  THEORY.  [Cn.  Hi. 

the  eccentricity  0.5  and  1%  of  steel  the  diagram  gives  M/bh*fc  =  0.171; 

hence 

1,195,500 


The  intersection  of  the  1%  curve  and  the  0.5  eccentricity  line  lies  to 
the  left  of  the  curve  fa/fc  •*•  15  ;  hence  the  unit  stress  in  the  tensile  steel 
is  less  than  15X647  =  9470  lbs/in2.  The  exact  value  can  be  computed 
from  equation  13;  the  value  of  k  for  use  in  it  can  be  obtained  easiest 
from  the  diagram  on  page  98 

(3)  The  breadth  of  a  beam  is  12  in.  and  its  height  24  in.    At  a  certain 
section  the  bending  moment  is  450,000  in-lbs.,  and  the  eccentric  distance 
is  4  in.      The  working  strength  of  the  concrete  being  600  lbs/in',  how 
much  steel  reinforcement,  if  any,  is  required? 

Solution.    The   eccentricity  is  e/h==^,  and  hence   the  beam  would 
be  on  the  border  between  Case  I  and  II  even  if  no  steel  were  used.    With 
steel,  the  beam  falls  under  Case  I,  and 
M  450,000 

&^/ri2X24*X600  =  °-10 

Entering  the  diagram,  page  287,  with  this  value  and  tracing  horizontally 
to  the  0.167  eccentricity  vertical,  we  find  their  intersection  and  note 
that  it  falls  between  the  0.6  and  0.8%  curves;  about  0.7%  of  steel  there- 
fore is  required. 

(4)  In  example   (3)  change   the   eccentric  distance  to  12  in.  and 
solve. 

Solution.  The  eccentricity  is  e/h  =  %,  and  the  beam  falls  under 
Case  II  (see  page  288).  M/bh2fc  has  the  same  value  as  in  example  (3); 
hence  entering  the  diagram  with  that  value  and  tracing  horizontally 
to  the  0.5  eccentricity  vertical,  we  find  their  intersection  and  note  that 
it  falls  between  the  0.2  and  0.3%  curves;  hence  0.3%  is  the  required 
amount 

(At  first  thought  it  may  seem  that  more  steel  is  necessary  in  example 
(4)  than  in  (3)  because  of  the  greater  eccentricity  in  the  former  example. 
But  it  should  be  noted  that  the  thrust  N  is  much  less  in  (4)  than  in  (3), 
its  values  being  M/e  =  37,500  and  112,500  Ibs,  respectively.) 

86.  Shearing  Stresses  in  Reinforced  Beams.  —  In  Art.  46 
the  variation  in  shearing  stress  in  a  homogeneous  beam  was 
discussed  and  the  general  formula  given  for  the  intensity  of 
shear  at  any  point  (see  eq.  (1)).  In  a  reinforced  beam  "the 
variation  in  shear  differs  from  that  in  a  homogeneous  beam 
owing  to  the  concentration  of  tensile  stress  in  the  steel.  The 


§86.] 


FLEXURE  AND  DIRECT  STRESS. 


109 


general  formula  for  shearing  stress  may,  however,  still  be  used 
if  the  transformed  section  be  employed;  that  is,  if  the  area 
of  the  steel  be  multiplied  by  n  and  considered  equivalent  to 
concrete  at  the  same  horizontal  plane.  The  tension  area  of 
the  concrete  should  be  neglected.  A  simpler  method  for 
present  purposes  is  the  following:  In  Fig.  36  is  represented  a 
short  portion  of  a  beam  where  the  total  vertical  shear  is  V. 
Let  v= horizontal  (^>r  vertical)  shearing  stress  per  unit  area 
at  the  neutral  axis,' and  let  b=  width  of  beam.  Other  quan- 
tities are  sufficiently  indicated  in  the  figure.  C=T  and  Cf  =  T'. 


Neutral  Plane 


_  Steel 


FIG.  37. 


The  total  shearing  stress  on  any  horizontal  plane  between  the 
steel  and  the  neutral  axis  will  be  equal  to  T'—T  and  the  stress 

T'-T 
per    unit    area=^=    ,„    .    From   equality  of  moments  we 

have  the  relation  Vdl=  (T'—T)]d,  whence  is  derived  the  expres- 
sion 


The  shearing  stress  given  by  eq.  (1)  is  the  same  at  all  points 
between  the  neutral  axis  and  the  steel;  above  the  neutral  axis 
the  shear  follows  the  parabolic  law  as  in  a  homogeneous  beam. 
Fig.  37  represents  the  law  of  variation  for  the  case  under 
discussion. 

Using  7/8  for  an  approximate  value  of  j  (see  Art.  55)  we 
have  approximately 

8   V 


v    7  bd' 


(2) 


110  GENERAL  THEORY  [Ca.  111. 

i 
that  is,  the  shearing  stress  at  the  neutral  axis  (equal  to  the 

maximum)  is  one-seventh  or  about  14%  more  than  the  aver- 
age value. 

87.  Beams  Reinforced  for  Compression.  —  In  beams  reinforced 
for  compression  formula  (1)  will  still  apply,  the  value  of  jd 
being  the  distance  between  the  tensile  steel  and  the  resultant 
of  the  compressive  stresses  as  shown  in  Art.  77.      In  this  case 
j  is  somewhat  greater  than  in  the  previous  case  and  v  is  more 

V 
nearly  equal  to  the  average  shearing  stress  7-7. 

88.  T  '-beams.  —  Here  again  formula  (1)  still  holds  true,   / 
retaining  its  general  significance.    As  shown  in  Art.  73,  /  may 
be  taken  as  closely  equal  to  the  distance  from  the  steel  to 
the  centre  of  the  flanges  of  the  T\    hence 


It  is  to  be  noted  that  in  the  T-beam  the  shearing  stresses 
are  practically  the  same  as  in  a  rectangular  beam  of  the  same 
depth  and  having  the  same  width  as  the  stem  of  the  T.  The 
slab  aids  in  reducing  the  shear  only  by  its  effect  in  increasing 
slightly  the  value  of  j. 

89.  Working  Formula.  —  Since  the  value  of  /  varies  only 
within  narrow  limits  it  is  quite  as  satisfactory  for  comparative 
purposes  and  for  purposes  of  design  to  use  the  average  value 
of  the  shearing  stress, 


in  which  6.  is  the  breadth  and  d  is  the  net  depth  of  the  beam. 
In  T-beams  b  is  the  breadth  of  the  s£gm,  and  d  is  the  total 
depth  from  top  of  beam  to  steel.  The  true  maximum  shear 
will  generally  be  from  10  to  15  per  cent  higher  than  the  aver- 
age value  thus  determined. 

90.  Effect  of  Shear  on  the  Tensile  Stresses  in  the  Concrete.  —  In 
Art.  46  it  was  shown  that  in  a  homogeneous  beam  the  direc- 


§90.]  SHEARING   STRESS.  Ill 

tion  of  the  maximum  tensile  stresses  is  horizontal  at  the  lower 
face  and  becomes  more  and  more  inclined  as  the  neutral  axis 
is  approached,  reaching  an  inclination  of  45°  at  that  places 
In  the  reinforced  beam  we  have  assumed,  for  purposes  of 
design,  that  there  is  no  tension  in  the  concrete.  WTiile  such 
possible  tension  will  add  very  little  to  the  resisting  moment 
of  the  beam  it  is  desirable  to  consider  it  here  in  relation  to 
the  shearing  stresses  and  the  resultant  effect  on  lines  of  prob- 
able rupture.  The  shearing  stresses  determined  in  the  pre- 
ceding article  have  been  calculated  on  the  assumption  of  no 
tensile  stress  in  the  concrete,  but  the  effect  of  such  tension 
on  the  distribution  of  the  shear  is  very  small  and  need  not  be 
considered. 

To  determine  the  amount  and  direction  of  the  maximum 
inclined  tensile  stresses  at  any  point,  eq.  (1),  Art.  46,  is  still 
applicable.  In  this  case  large  shearing  stresses  exist  imme- 
diately above  the  steel,  hence  the  maximum  tensile  stresses 
become  considerably  inclined  as  soon  as  we  leave  the  line  of 
the  steel,  the  exact  direction  depending  upon  the  relation 
between  the  shear  and  the  horizontal  tension.  Exact  calcula- 
tions are  impossible,  since  the  actual  horizontal  tension  in  the 
concrete  is  unknown.  While  the  steel  is  assumed  to  carry  all 
tension  the  concrete  will  in  fact  be  stressed  in  accordance 
with  its  deformation  up  to  the  point  of  ultimate  deformation 
and  rupture.  Where  the  steel  has  a  stress  of  its  full  working 
value  of  12,000  to  15,000  Ibs/in2,  the  deformation  will  mucn 
exceed  the  ultimate  deformation  of  the  concrete  and  rupture 
must  occur,  but  at  points  where  the  steel  stress  is  low,  as  for 
example  near  the  end  of  the  beam,  the  concrete  may  be  intact. 

Suppose,  for  example,  the  stress  in  the  steel  is  3000 
Ibs/in2.  If  the  modulus  of  elasticity  of  the  concrete  hi  ten- 
sion is  1,500,000  the  stress  in  it  will  be  3000/20  =  150  Ibs/in2, 
which  is  not  far  from  its  ultimate  strength.  Suppose  further 
that  the  unit  shearing  stress  in  the  lower  part  of  the  beam  is 
100  Ibs/in2.  By  eq.  (2)  of  Art.  46  the  resultant  maximum 
tension  will  be  £=i(150)+v/i-1502+1002=200  Ibs/in2,  and 

>t\^~ 


112  GENERAL  THEORY.  [Cn.  III. 

will  have  a  direction  inclined  26|°  from  the  horizontal.  This 
stress  may  exceed  the  ultimate  strength  of  the  concrete  and 
the  result  will  be  an  inclined  crack.  At  points  nearer  the 
neutral  axis  the  horizontal  tensile  stresses  become  less  and 
^~»  the  inclined  tension  approaches  the  value  of  the  shearing 
stress  and  its  inclination  approaches  45°.  The  result  of  these 
inclined  stresses  is  likely  to  be  a  progressive  tension  failure 
in  an  inclined  direction  which  the  horizontal  rods  are  not  very 
effective  in  preventing. 

Excessive  stresses  of  this  kind  are  prevented  in  various 
ways.  Obviously  they  will  be  reduced  by  keeping  the  hori- 
zontal tension  small  through  the  use  of  considerable  horizontal 
steel  at  points  of  heavy  shear,  by  keeping  the  shearing  stresses 
low,  and  by  various  means  of  directly  carrying  the  inclined 
stresses  by  special  reinforcement. 

91.  Ratio  of  Length  to  Depth  for  Equal  Strength  in  Moment 
and  Shear.  —  For  any  given  values  of  per  cent  of  steel  and  of 
working  stresses  in  shear  and  direct  stress  there  is  a  definite 
ratio  of  length  to  depth  of  beam  which  will  give  equal  strength 
in  moment  and  shear.  The  strength  of  beams  of  greater  rela- 
tive Length  will  be  determined  by  their  moment  of  resistance, 
while  that  of  shorter  beams  by  their  shearing  resistance.  The 
ratio  of  length  to  depth  for  equal  strength  depends  on  the 
method  of  loading. 

For  Single  Concentrated  Loads.  —  In  this  case  the  shear  V,  due 
to  a  given  load  W,  is  %W,  and  the  moment  M  is  \WL  Hence 

W  =  2V  =  4M/l.     .   ',    .    .    .  ..     (a) 

From  Art.  89  we  have  V  =  v'bd  and  from  Art.  56 
M8  =  fajpbd2,  in  which  i/  =  safe  average  shearing  stress  and 
}8  =  working  stress  in  steel.  Substituting,  we  have 


from  which 

LJ 

d       v' 


§93.]  SHEAPtlNG  STRESS.  113 

For  a  Uniformly  Distributed  Load  a  similar  process  gives  the 
ratio 


For  Beams  Loaded  with  Equal  Loads  at  the  Third  Points, 

' 


d        vf 


» 


(3) 


In  the  case  of  continuous  girders  these  formulas  will  apply 
if  I  be  taken  as  the  length  between  points  of  inflection. 

Taking,  for  example,  p  =  0.01,  ?/  =  50  lbs/in2,  and  /,=  15,000 
lbs/in2,  and  using  an  average  value  of  7/8  for  /,  we  have  the 

following  ratios  for  -r: 

For  concentrated  loads  -7-  =5.25. 

For  uniformly  distributed  loads  j  =  10.5. 

a  « 

For  double  concentrated  loads  -7  =  7.87. 

a 

92.  Bond  Stress.  —  The  stress  on  the  bond  between  steel 
and  concrete  (Fig.  36,  Art.  86)  will  be  equal  to  T'-T  on 
the  length  dl. 

If  U  denote  the  bond  stress  per  lineal  inch,  we  then  have 

Tf  —  T 

TT_  _  _  f_ 

u  —       T;  —  . 
dl 

whence  we  derive 


The  bond  stress  per  unit  area  will  be  equal  to  U  divided  by 
the  sum  of  the  perimeters  of  the  steel  sections.  Or,  if  o  = 
perimeter  of  one  bar,  So  =  sum  of  perimeters  and  u  =  bond  stress 
per  unit  area,  we  have 

V 

II  --  fO> 

Zo-jd  ........     W 


114  GENERAL   THEORY.  [Cn.  Ill 

g.2a.  Bond  Stress  for  Compressive  Reinforcement.  —  The  ques- 
tion of  bond  stress  for  compressive  reinforcement  will  seldom 
come  into  consideration.  In  this  case  it  may  also  be  calcu- 
lated by  formulas  based  on  the  shear  at  the  section,  but  it  is 
necessary  to  take  account  of  the  compression  carried  by  the 
concrete,  and  the  formula  is  the  general  formula  for  horizontal 
shear,  involving  the  statical  moment  of  the  concrete  area  and 
the  equivalent  steel  area  about  the  neutral  axis. 

By  this  general  formula  the  horizontal  shear  per  lineal  inch 
at  any  horizontal  plane  is  proportional  to  the  statical  moment 
of  the  effective  area  outside  such  plane  about  the  neutral  axis. 
At  the  neutral  axis  this  shear  per  lineal  inch  has  been  shown 
to  be  vb=V/jd.  The  shear  between  the  compression  rods  and 
the  concrete,  or  the  bond  stress  desired,  will  therefore  be  equal 

V     Statical  moment  of  equivalent  steel  area 
to  -,X^-       -r-  V"  -.    The    total 

yd    Total  statical  moment  of  compression  area 

moment  of  the  compression  area  equals  the  total  moment  of 
the  tension  area,  which  is  nA(l  —  k)d,  where  A  is  the  total  area 
of  tensile  steel.  (See  Art.  76  for  notation.)  The  moment  of 
the  compressive  steel  is  equal  to  nA'(kd~  df).  Hence,  for  bond 
stress  of  compressive  steel, 

V     A'(kd-d') 
~X 


That  is,  the  bond  stress  per  lineal  inch  for  the  two  steel  areas 
will  be  proportional  to  the  areas  times  their  distances  from 
the  neutral  axis.  Since  the  compressive  steel  will  generally 
be  nearer  the  neutral  axis  than  the  tensile  steel  it  follows  that 
if  the  compression  bars  are  no  larger  in  diameter  than  the 
tension  bars,  the  bond  stress  will  always  be  less  than  that  in 
the  tension  bars. 

926.  Variation  of  Bond  Stress  in  Beams.  —  The  theoretical 
results  assume  that  there  is  perfect  adhesion  and  that  the 
stress  in  the  steel  is  taken  over  by  the  concrete  at  all  points, 
as  called  for  by  the  usual'theory.  They  show  that  the  bond 


§  926.] 


BOND    STRESS. 


115 


stress  is  a  simple  function  of  the  shear  and  varies  therewith. 
Thus,  in  a  beam  supporting  a  single  concentrated  load,  the 
shear  and  bond  stress  varies,  as  shown  in  Fig.  37a.  In  this 
case  the  bond  stress  is  uniform  from  load  to  end  of  beam.  For 
a  distributed  load  the  variation  is  as^  shown  in  Fig.  376.  The 
value  is  a  maximum  at  the  ends  and  decreases  towards  the 
centre.  Thus  if  a  reinforcing  rod  carries  a  certain  stress  S  at 

I 


FIG.  37a. 


FIG.  376. 


the  centre,  in  the  first  case  the  bond  stress  per  lineal  inch  will 
be  uniform  and  may  be  calculated  by  the  formula  U  =  S-t--. 


Or,  by  eq.   (1),  it  is 


=--=.      But    S  =      =.   Hence 
]d     ]d  ]d     ]d 


U*=S-*--t  as  above  given.     In  the  second  case  the  shear  and 

bond  stress  is  not  uniform  and  hence  the  bond  stress  is  not 
to  be  calculated  by  dividing  the  stress  8    § 
by  1/2.     It  is  a  maximum   at   the  end 
where  it  is  equal  to   twice   the   average 
value.     In    practice,  a    beam  will  jhave 
some  length  beyond  the  theoretical  centre 
of  bearing,   and    the    rods   will   extend 
entirely  through  the  beam.     This  doubt- 
less modifies  the  bond  stress   near   the 
end,  somewhat  as  shown  by  the  dotted 
lines  in  Fig.  37c,  some  stress  being  carried  by  the  rod  beyond  the 
theoretical  centre  of  bearing,  thus  reducing  the  maximum  bond 
stress  below  the  theoretical  value.     In  the  case  of  continuous 


FlG  37c 


116  GENERAL    THEORY.  [CH.  III. 

girders  the  bond  stress  is  still  a  maximum  at  the  support  and 
is  measured  by  the  shear,  although  the  stress  may  be  corn- 
pressive  (the  increment  of  stress  is  still  of  the  same  sign).  Con- 
tinuous rods  would  tend  to  modify  the  bond  stress,  as  shown 
in  Fig.  576,  Art.  123,  thus  reducing  the  maximum  on  both 
sides  of  the  support,  and  the  maximum  compression  in  the 
steel.  The  stress  in  the  concrete  will  be  increased.  For  dis- 
cussion of  anchored  rods  see  Art.  123. 

92c.  Deflection  of  Reinforced  Concrete  Beams. — Deflection 
formulas  for  homogeneous  beams  can  be  interpreted  semi- 
rationally  to  make  them  applicable  to  reinforced  concrete  beams. 
So  interpreted  they  yield  results  in  fair  agreement  with  actual 
measured  deflections.  (See  Art.  112a.) 

General  Theory. — As  is  well  known,  a  concrete-steel  beam 
under  full  working  load  contains  one  or  more  cracks  at  or  near 
the  section  of  maximum  bending  moment  or  else  the  condition 
there  is  near  the  cracking  stage ;  and  to  compute  the  maxi- 
mum unit  fibre  stresses  at  such  section,  engineers  rightly 
assume  the  presence  of  a  tension  crack,  and,  in  effect,  that 
it  has  extended  to  the  neutral  axis.  Since  the  deflection 
depends  on  the  stress  at  all  sections,  and  the  cracked  sections 
are  comparatively  very  few,  a  deflection  formula  should  be 
based  on  the  intact  section.  It  may  be  thought  that  a  cracked 
section  influences  the  deflection  more  than  an  intact  one,  the 
idea  is  correct,  but  the  effect  of  incipient  cracking  on  the 
deflection  is  not  as  great  as  on  the  fibre  stress  at  the  section. 
These  effects  are  entirely  different  in  "order  of  magnitude," 
the  first  is  not  noticeable  at  all  in  careful  measurements  on 
deflections  due  to  increasing  loads,  whereas  the  latter  certainly 
would  be  if  fair  measurements  of  fibre  stress  at  a  section 
of  a  beam  were  possible.  To  simplify  certain  relations, 
it  will  be  assumed  that  the  depth  of  the  intact  section  for 
use  in  the  deflection  formula  extends  from  the  top  of  the  beam 
to  the  centre  of  the  steel,  this  in  effect  assumes  all  sections 
cracked  from  the  bottom  to  the  centre  of  the  steel. 

Deflection    formulas    for  homogeneous  beams  imply  that 


§  92c.]  DEFLECTION.  117 

the  material  of  the  beam  obeys  Hooke's  law  ("  stress  is  pro- 
portional to  strain"),  up  to  working  stresses  at  least,  and  that 
the  moduli  of  elasticity  of  the  material  for  tension  and  com- 
pression are  equal.  While  it  is  true  that  concrete  does  not 
obey  the  law  strictly,  still  its  stress-strain  relation  for  com- 
pression is  nearly  linear  up  to  working  stresses.  But  the  stress- 
strain  relation  for  tension  is  far  from  linear,  and  the  assumption 
that  it  is,  herein  made  for  simplicity  in  formulas,  must  be 
regarded  as  a  rough  approximation.  It  is  true  that  the  "initial 
moduli"  (Art.  24)  of  concrete  for  compression  and  tension  are 
nearly  equal,  but  the  deflection  of  a  beam  depends  on  the 
elongations  and  shortenings  of  all  the  fibres,  and  hence  not 
upon  initial  modulus  but  on  some  sort  of  a  mean  value.  This 
is  not  the  modulus  corresponding  to  the  mean  unit  fibre 
stress,  but  certainly  the  average  or  secant  modulus  is  nearer 
correct  than  the  initial  or  the  modulus  at  the  maximum  unit 
stress. 

The  formulas  also  imply  that  the  moments  of  inertia  of 
the  cross-sections  of  the  beam  are  equal.  This  condition  is 
not  fulfilled  in  most  reinforced  concrete-beams,  due  account 
being  taken  of  the  steel,  because  of  presence  of  bent-up  rods 
and  stirrups.  Still  the  amount  of  steel  in,  and  hence  the 
moments  of  inertia  of,  sections  in  the  middle  third  or  middle 
half  are  commonly  constant;  and  since  the  middle  half  con- 
tributes nearly  85%  of  the  maximum  deflection  in  the  case  of 
a  simple  beam  constant  in  section  and  uniformly  loaded,  and 
82%  when  the  beam  is  loaded  at  the  two  outer  points,  it  must 
be  that  a  small  change  in  the  moments  of  inertia  of  end  sec- 
tions of  a  simple  beam  would  produce  a  much  smaller  change 
in  the  maximum  deflection.  In  fact,  if  a  simple  beam  is  uni- 
formly loaded,  for  example,  and  the  moment  of  inertia  of 
sections  in  its  middle  half  is  /i,  and  that  of  sections  in  its 
outer  quarters  is  /2,  then  its  maximum  deflection  is 
TF/3(67/2  +  13/i)/6144£J/i72;  and  if  the  sections  are  uniform 
and  the  common  moment  of  inertia  is  /i,  then  the  maximum 
deflection  is  5WP/EIi  384,  hence  the  ratio  of  the  deflections  is 


118  GENERAL    THEORY.  [CH.  III. 

(67/2  +  13/i)/80/2,  and  if  h  and  72  differ  by  10%,  say,  the 
maximum  deflections  differ  by  less  than  2%. 

For  the  reasons  stated  above,  the  deflection  formulas  for 
homogeneous  beams  will  be  used  for  reinforced-concrete  beams, 
but  modified  in  accordance  with  the  following  assumptions: 

1.  That  the  representative  or  mean  section  has  a  depth 

equal  to  the  distance  from  the  top  of  the  beam  to  the 
centre  of  the  steel; 

2.  That  it  sustains  tension  as  well  as  compression,  both 

following  the  linear  law; 

3.  That  the  proper  mean  modulus  of  elasticity  of  the  con- 

crete equals  the  average  or  secant  modulus  up  to  the 
working  compressive  stress;  and 

4.  That  the  allowance  for  steel  in  computing  the  moment  of 

inertia  of  the  mean  section  should  be  based  on  the 

amount  of  steel  in  the  mid-sections. 

92cL  Deflection  of  Rectangular  Beam,s.—Yor  homogeneous 
beams  the  deflection  formulas  commonly  involve  the  load  or 
the  maximum  unit  fibre  stress.  The  following  are  correspond- 
ing formulas  for  rectangular  reinforced  concrete  beams: 


and 


?.  d 
or 


In  these  the  notation  is  as  follows : 

D  =  maximum  deflection   (if  desired  in  inches,  the  units 

specified  below  should  be  used) ; 
b  =  breadth  of  the  section  (in.); 
d  =  depth  of  the  section  to  the  centre  of  the  steel  (in.); 


§  92d.]  DEFLECTION.  119 

Ci=the  numerical  coefficient  in  the  formula  for  deflection  of 
homogeneous  beams,  CiWl3/EI,  depending  on  the 
loading  and  support  (see  page  126); 

C2=the  numerical  coefficient  in  the  formula  for  maximum 
bending  moment,  c^Wl,  also  depending  on  the  loading 
and  support  (see  page  126) ; 
Es  =  modulus  of  elasticity  of  the  reinforcing  steel  (lbs/in2); 

E  =  modulus  of  elasticity  of  the  concrete  (lbs/in2); 

n= ratio  of  the  moduli  of  elasticity  of  steel  and  concrete; 

p  =  steel  ratio  (area  of  steel  section  -f-  bd) ; 

a  =  a  numerical  coefficient  depending  on  p  and  n\ 

fc  =  greatest  unit  compressive  stress  in  the  concrete 
(lbs/in2); 

fs  =  greatest  unit  tensile  stress  in  the  steel  (lbs/in2); 

A;  =  proportionate  depth  of  the  neutral  axis  (see  Fig.  16); 

y= proportionate  distance  of  the  centroid  of  the  compres- 
sive stress  from  the  steel  (see  Fig.  16). 

The  schedule  (page  126)  gives  values  of  ci/Es  for  use  in  for- 
mula (1)  and  values  of  ci/c2Es  for  formulas  (2)  and  (3)  for  cer- 
tain standard  cases  more  or  less  close  approximations  to  which 
are  met  in  practice;  and  the  diagrams  (Figs.  37/  and  37g) 
furnish  values  of  n/a,  kj,  and  pj.  It  is  recommended  that 
8  or  10  be  used  for  n  in  the  first  diagram  (see  Art.  112c);  in 
the  second  that  value  of  n  is  to  be  used  which  the  computer 
prefers  in  his  own  formulas,  tables,  or  diagrams  for  the  strength 
of  beams.  These  two  values  of  n  will  probably  be  unlike; 
the  apparent  inconsistency  is  discussed  at  the  close  of  this 
article. 

Example  1. — A  concrete  beam  rests  on  end  supports  16  ft.  apart, 
the  breadth  of  its  section  is  10  in.,  the  depth  (to  the  steel)  is  15  in., 
the  reinforcement  consists  of  four  f-in.  rods  extending  along  the  whole 
length  (and  stirrups).  What  is  its  probable  deflection  when  sustain- 
ing a  uniform  load  of  10,000  Ibs.,  including  its  own  weight? 

The  amount  of  steel  is  1.767  in2,  hence  p  =  1.767  ^-150  =  . 012. 
Entering  the  diagram  (page  124)  at  percentage  1.2,  tracing  upward 
to  the  n  =  S  curve  say,  and  then  horizontally,  it  is  found  that  n/a  =  76. 


120  GENERAL   THEORY.  [Cn.  IIL 

From  the  schedule  (page  126)  it  is  found  that  cjEs  =  0.000434/1,000,000. 
hence  from  eq.  (1) 

.000434  X  10,000  X  1923  X  76  . 

1,000,OOOX10X153 

Example  2.—  The  deflection  of  the  beam  described  in  the  preceding 
example  is  desired,  (1)  when  it  is  loaded  so  that  the  working  com- 
pressive  fibre  stress  is  500  lbs/in2,  and  (2)  when  the  working  stress  in 
the  steel  is  14,000  lbs/in2. 

(1)  From  the  schedule  it  is  seen  that  cJc2Es  =  .  00347  /1,  000,000,  and, 
as    in    example    1,    n/«  =  76.     Entering    the    diagram    (page    125)   at 
p  =  1.2%  and  tracing  upwards  to  the  n  =  l5kj  curve  (a  value  of  n  much 
used  in  strength  formulas),  and  then  horizontally  to  the  left,  we  find 
that  kj  is  .38;  hence  from  eq.  (2), 

^     .00347  X  500  X1922X.  38X76 
2X1,000,000X15 

(2)  Entering  the  diagram  at  p  =  1.2%  and  tracing  upward  to  the 
n  =  15pj  curve  and  then  horizontally  to  the  right  we  find  that  p/  =  .0102; 
hence  from  eq.  (3) 

.00347  X  14,000  X  192  2X.  0102X76  . 

1,000,000X15 

Analysis  for  Formulas  and  Diagrams.  —  Since  the  total 
tension  (in  concrete  and  steel)  and  the  total  compression  are 
equal  (see  Fig.  37d),  at  any  section, 


A  n 

Also  /s  /  ft  =/c  (!  —  &)/  A;  and  A=pbd,  and  these  values  substituted 
in  the  first  equation  yield  one  from  which  it  follows  that 


The  moment  of  inertia,  with  respect  to  the  neutral  axis,  of  the 
part  of  section  in  compression  is  %bk3d3,  that  of  the  concrete 


§  92d.] 


DEFLECTION. 


121 


section  in  tension  is  %b(l-k)Bd3,  and  that  of  the  weighted  steel 
sections  is  practically  nA(l  —  k)2d2;  hence 


or 

and 


-W  +  3np(l-k)2],      ....     (5) 
D=ciWl3/EcI=ClWPnlE8bd*a, 


which  is  eq.  (1). 

From  eqs.  (4)  and  (5),  the  value  of  a  for  any  values  of  p 


FIG.  37<*. 

and  n  may  be  computed;  a  sufficient  number  of  these  were 
thus  computed  to  determine  the  n/a  curves  in  Fig.  37 f. 

The  transformation  of  the  deflection  formula  (1)  (in  terms 
of  the  load)  into  (2)  and  (3)  (in  terms  of  the  working  unit 
stresses  fc  and  fs  respectively)  will  now  be  made.  For  this 
purpose,  strength  formulas  based  on  cracked  sections  (see 
Fig.  16)  and  a  linear  variation  of  compression  are  used.  These 
well-known  strength  formulas  based  on  concrete  and  steel  are 
respectively,  M  =  %fckjbd2  and  M=fspjbd2  (see  page  56). 
Since  M=c2Wl  also,  W  =  %fckjbd2/c2l=f*pibd2/C2L  These  two 
values  of  W  substituted  in  eq.  (1)  yield  eqs.  (2)  and  (3)  re- 
spectively. 

The  formulas  for  k  and  j  of  Fig.  16  are  also  well  known; 
they  are  (see  Art.  55) 


k=V2pn+(pri)2-pn (6) 


122 


GENERAL    THEORY. 


[OH.  III. 


and 

A i i  if  (n\ 

j  —  1  —  3^/C.      .       »       .       .       .       .       .       .       {f ) 

By  means  of  these  the  values  of  kj  and  pj  can  be  computed 
for  any  values  of  p  and  n',  a  sufficient  number  of  these  were 
thus  computed  to  determine  the  kj  and  pj  curves  of  Fig.  370. 

Choice  of  different  values  of  n  in  n/a  and  kj  or  pj  for  use 
in  any  particular  case  is  not  an  inconsistency.  The  first  value 
depends  on  the  unit  fibre  stresses  at  all  points  of  the  beam, 
and  when  the  numerical  value  is  chosen  from  experiments  on 
deflection,  then  n  becomes  also  a  sort  of  empirical  coefficient- 


FIQ.  37e. 

making  correction  for  various  errors  in  the  deduction  of  the  deflec- 
tion formula,  whereas  the  second  depends  on  the  unit  stresses  in 
the  cracked  section  and  when  its  numerical  value  is  chosen 
from  experiments  on  the  strength  of  beams,  then  it  also  becomes 
in  part  an  empirical  coefficient  correcting  errors  of  approxima- 
tion in  the  strength  formulas  used. 

926.  Deflection  of  T-beams. — Under  the  four  assumptions 
stated  in  Art.  92c,  the  deflection  formula  for  T-beams  in  terms 
of  the  load  becomes 


_ci  W I3  n 
¥  bd*     ' 


(1) 


in  which  p  is  a  coefficient  depending  upon  the  steel  ratio  and 
n,  b  width  of  flange,  d  depth  to  steel  (see  Fig.  27);  other 
symbols  are  explained  in  Art.  92d. 


§  92e.]  DEFLECTION.  123 

In  accordance  with  assumption  2,  the  neutral  axis  of  the 
representative  section  will  be  in  the  web,  or  stem,  generally, 
as  is  implied  in  Fig.  37e.  Then  the  total  tension  and  com- 
pression at  the  section  are  given  by 

T=b'(l-k)d%fc(l-k)/k+pbdnfc(l-k)/k 
and  C 


Since  T=C,  their  values  may  be  equated;  the  resulting  equa- 
tion leads  to 


r&       /t\2    /r\2i 

»+»[>  -ik)  +G-)  ] 


*-  V    Vt     t      ~  .....    (2) 

,     ,         np+____+_     1 

The  moment  of  inertia  of  the  concrete-steel  section,  the  steel 
area  being  weighted  n-fold,  is  given  by 


and  if  /?  be  used  to  denote  this  coefficient  of  bd3,  then 

-k.     (3) 


Example.  —  A  T-beam  rests  on  end  supports  10  ft.  apart  and  sustains 
loads  of  5000  Ibs.  at  its  third  points.  f  The  dimensions  of  the  section 
are  6  =  16  in.,  b'=S  in.,  d  =  10  in.,  arid  t=3%  in.;  and  the  reinforcement 
consists  of  three  f-in.  square  bars.''  What  is  the  probable  deflection  due 
to  the  load? 

Solution.  The  steel  ratio  is  .011;  and  with  n=8,  eq.  (2)  gives 
/b  =  .485,  and  eq.  ~(3)  gives  /?  =  .0835.  Now  for  loads  at  third  points, 
Ci  =  23/1296;  hence 

23  10,OOOX1203  8 

' 


1296  30,000,000  X  16  X103  0.0835 


124 

180 
170 
160 
150 
140 
f  130 

I  12° 
1 

100 
90 
80 
70 
60 

GENERAL    THEORY.                            [Cn.  III. 

Ci    WZ  3  / 

a* 

D^  Es    bd*( 

n 

=15 

\ 

\ 

\ 

\ 

\ 

\ 

n 

=12 

\ 

\ 

\ 

\ 

s 

\ 

\ 

^ 

\ 

\ 

X 

x 

n 

=10 

N 

x 

\ 

ss 

X 

x 

^ 

x 

^ 

^ 

^^ 

\ 

x 

^ 

\ 

^ 

\ 

^ 

^ 

n 

=  8 

^ 

<; 

s 

< 

^ 

^ 

^ 

^. 

^- 

s^ 

^--. 

--^ 

"~--^ 

•  ^ 

—  ^, 

•^^. 

-^^ 

-^. 

•—  ^ 

•~^~. 

—  »_ 

•—  —  . 

"—  -^ 

0.5  1.0  1.5 

Percentage-of  Reinforcement 


FIG.  37/. 


§  92e.] 


DEFLECTION. 


125 


45 


.40 


35 


.30 


o 
£.25 


.20 


.15 


|  =  15  / 


D_l  £i_ 

D~2    C2E« 


7 


/ 


.015 


.010 


Ci          /s£2 

d 


.005 


.004 


0.5  1.0  1.5 

Percentage  of  Reinforcement 

FIG.  37^. 


126 


GENERAL    THEORY. 


[Cn.  III. 


SCHEDULE  OF  COEFFICIENTS. 


D-c,WP/El 

M  =  c2Wl 

<k 

C2 

In  millionths. 

ci 
*^ 

Ci 
*Ct&* 

F                p; 

i 

i 

-A 

U   "4 

.00932 
.0054 

T^a' 

i 

1 

.0111 
.00417 

.000094 
.000434 

.000301 
.000180 

.000173 
.000087 

.0111 
.00834 

.00278 
.00347 

.ooieo 

.00144 

.00139 
.00210 

E5 

W             pp 

1 

jW 

2 

i 
i 

A                                     A 

"w 

A 

r      * 

*^           ^f 

•w               , 

-1% 
1 

±n 

H? 
'  fl                F             p: 

±1 
-A 

*"               "~E 

H                  W                  -h- 

1  1                                         K1 

^              ^S. 

384 

*  For  £s  =  30,000,000  lbs/in2  ir  this  schedule. 


93.  Strength  of  Columns. — Concrete  columns  need  rarely 
be  calculated  as  long  columns.  In  ordinary  construction 
the  ratio  of  length  to  least  width  will  seldom  exceed  12  or  15, 
while  the  results  of  tests  indicate  little  or  no  difference  in 


§  95.J  STRENGTH  OF  COLUMNS.  127 

strength  for  ratios  up  to  20  or  25.  It  will  be  desirable  then 
to  determine  first  the  strength  of  a  reinforced  column  con- 
sidered as  a  short  column.  If  the  conditions  require  it  a  gen- 
eral column  formula  may  then  be  applied  to  provide  for  cases 
where  the  length  is  excessive. 

94.  Methods  of  Reinforcement. — Columns  are  reinforced  in 
two  ways:  (1)  by  means  of  longitudinal  rods  extending  the 
full  length  of  the  column,  and  (2)  by  means  of  bands  or  spirally 
wound  metal.  In  the  first  case  the  steel  aids  by  carrying  a 
part  of  the  load  directly,  the  stresses  in  the  two  materials 
being  proportional  to  their  moduli  of  elasticity.  In  the  other 
case  the  steel  supports  the  concrete  laterally,  preventing  lateral 
expansion  to  a  greater  or  less  degree,  and  thus  strengthening 
the  concrete.  Usually  both  methods  are  more  or  less  com- 
bined, the  longitudinal  rods  being  frequently  bound  together 
at  intervals  by  circumferential  bands  of  some  sort,  and  on 
the  other  hand  hoops  or  spiral  wire  being  conveniently  held 
in  place  by  longitudinal  rods.  Experiments  show  that  both 
types  of  reinforcement  are  effective  in  raising  the  ultimate 
strength  of  a  column,  but  conclusive  results  have  not  been 
reached  as  to  the  true  relative  effect  of  different  types  and 
amounts  of  reinforcement. 

95'.  Columns  with  Longitudinal  Reinforcement. — As  long  as 
the  steel  and  concrete  adhere  the  relative  intensities  of  stress 
in  the  two  materials  will  be  as  their  moduli  of  elasticity,  using 
the  modulus  as  explained  in  Art.  24. 

Let  A  denote  total  cross-section  of  column; 

Ac     "     cross-section  of  concrete;  <. 

A3     "      cross-section  of  steel; 

p       "      ratio  of  steel  area  to  total  a,Yea,=A8/A; 

fc       "      stress  in  concrete; 

n       "      ratio  of  moduli  of  steel  and  concrete  at  the 
given  stress  fc,=Es/Ec; 

P      "     total  strength  of  a  plain  column  for  the  stress  /c; 

Pf     "     total  strength  of  a  reinforced  column  for  the 


128  GENERAL    THEORY.  [Cn.  III. 

Then  P=fcA    .    ......    .    e     (a) 


and  P'=jc 

whence  P'=fcA[l  +  (n-l)p],    ........     (1) 

from  which  also 


(2) 


The  relative  increase  in  strength  caused  by  the  reinforce- 
ment is 

P'-P 

-    —  (n-l)p  .......     (3) 


The  elastic  limit  of  the  steel,  if  low,  may  affect  the  ulti- 
mate strength  of  the  column.    The  value  of  Pr  is  then 


(4) 


in  which  fei  is  the   elastic  limit  strength   of  the  steel.     (See 
Chapter  IV  for  further  discussion  of  this  question.) 

Eq.  (2)  is  convenient  to  use  in  determining  the  relative 
strength  of  a  reinforced  as  compared  to  a  plain  concrete  col- 
umn for  a  given  percentage  of  steel.  Thus  if  p  =  l%  and 

P' 

n  =  15,  we  have  p-  =  1  +  0.  14  =  1.14.     Thus   a  reinforcement  of 

1%  increases  the  strength  by  14%. 

From  these  relations  it  is  seen  that  the  relative  increase 
in  strength  caused  by  a  given  amount  of  reinforcement  depends 
on  the  value  of  n  and  is  greater  the  larger  n  is. 

The  economy  of  steel  reinforcement  is  also  dependent 
upon  the  working  stresses  permissible  in  the  concrete  since 


95.] 


STRENGTH    OF   COLUMNS. 


129 


fa^nfc.  The  following  table  shows  the  various  working  stresses 
in  the  steel  corresponding  to  various  values  of  working  stress 
in  the  concrete  and  to  various  values  of  the  modulus  Ec>  ther& 
is  given  also  the  percentage  increase  in  strength  for  each  one 
per  cent  of  steel. 


TABLE  No.  6. 
LONGITUDINAL  REINFORCEMENT  OF  COLUMNS. 


IbsVin.' 

EC, 

lbs/in2 

Ratio  of  Moduli, 
n 

f» 

lbs/in2 

Percentage  Increase 
in  Strength  for 
each  1  %  Rein- 
forcement. 

[ 

750,000 

40 

12,000 

o9 

orvn 

1,000,000 

30 

9,000 

29 

ouu 

1,500,000 

20 

6,000 

19 

I 

2,000,000 

15 

4,500 

14 

r 

1,000,000 

30 

12,000 

29 

400 

1,500,000 
2,000,000 

20 
15 

*8,000 
6,000 

19 
14 

I 

2,500,000 

12 

4,800 

11 

f 

1,000,000. 

30 

15,000 

29 

CATJ 

1,500,000 

20 

10,000 

19 

O\J\J              •( 

2,000,000 

15 

7,500 

14 

.  I 

2,500,000 

12 

6,000 

11 

r 

1,500,000 

20 

15,000 

19 

coo 

2,000,000 

15 

10,000 

14 

OUU              ^ 

2,500,000 

12 

7,200 

11 

I 

3,000,000 

10 

6,000 

9 

r 

2,000,000 

15 

12,000 

14 

Of\f\ 

2,500,000 

12 

9,600 

11 

o(J(J            •( 

3,000,000 

10 

8,000 

9 

[ 

3,500,000 

8.6 

6,900 

7.6 

From  this  table  the  relation  among  the  various  quantities 
may  be  clearly  appreciated.  It  is  to  be  noted  that  the  work- 
ing stresses  in  the  steel  must  be  relatively  low  except  in  the 
unusual  combination  of  high  working  stresses  in  the  concrete 
with  low  modulus.  High-grade  concrete,  permitting  high  work- 
ing stresses,  will  have  a  high  modulus.  For  further  discussion 
of  the  relations  of  working  stresses,  see  Chapter  V. 


130  GENERAL    THEORY.  [Cn.  III. 


Examples.  —  (1)  What  will  be  the  safe  strength  of  a  column  I 
in  cross-section  which  is  reinforced  with  1.5%  of  steel,  the  working  stress 
in  the  concrete  being  400  lbs/in2.    Take  n  =  15. 

From  eq.  (1)  wt,  have 

P'=400  X  15  X  15  X  (l  +  14  X~)  =90,000(1  +0.21)  -  108,900  Ibs. 

The  strength  of  the  plain  concrete  column  would  be  90,000  Ibs.,  and 
the  relative  increase  in  strength  is  21%.  The  stress  in  the  steel  would 
be  15X400  =  6000  lbs/in2. 

(2)  The  area  of  a  column  is  120  sq.  in.,  load  to  be  carried  is  60,000 
Ibs.,  and  working  stress  on  the  concrete  is  400  lbs/in2.  What  percentage 
of  steel  will  be  required?  Take  n  =  15. 

The  safe  strength  of  a  plain  concrete  column  would  be  120X400— 

P'     60 
48,000  Ibs.    Hence,  from  eq.  (2),  w  =  —  =  1  +  (15  -  l)p.    Hence 


96.  Columns  with  Hooped  Reinforcement.  —  Whenever  a 
material  subjected  to  compression  in  one  direction  is  restrained 
laterally,  then  lateral  compressive  stresses  are  developed  which 
tend  to  neutralize  the  effect  of  the  principal  compressive  stresses 
and  thus  to  increase  the  resistance  to  rupture.  Were  the  com- 
pressive stresses  equal  in  all  directions  there  would  be  no  rup- 
ture (as  there  would  be  no  shear).  The  strengthening  effect 
of  lateral  banding  depends  then  upon  the  rigidity  of  the 
bands,  that  is,  upon  the  amount  of  steel  used  and  its  closeness 
of  spacing.  Its  elastic  limit  may  also  affect  the  ultimate 
strength  of  the  column. 

On  the  basis  of  the  relative  lateral  and  horizontal  defor- 
mation of  the  concrete  (Poisson's  ratio)  it  is  possible  to  deduce 
a  theoretical  relation  between  the  lateral  and  the  longitudinal 
stresses,  and  thence  the  portion  of  the  longitudinal  stress 
remaining  unbalanced.  Let  /*= Poisson's  ratio,  fc= unbal- 
anced or  excess  of  longitudinal  over  lateral  compressive  unit 
stress, //  =  total  longitudinal  unit  stress,  /,  =  unit  tensile  stress 


§96.]  STRENGTH    OF    COLUMNS.  131 

in  steel,  p  =  steel  ratio  (reinforcement  to  be  closely  spaced). 
We  find  approximately 


.......     (1) 

and 

•     .     (2) 


Recent  experiments  by  Talbot  indicate  that  Poisson's  ratio 
for  concrete  is  quite  small,  probably  not  greater  than  rV  or  J. 

*  Demonstration.  (See  Johnson's  "Materials  of  Construction".)  —  Let 
/<=  Poisson's  ratio;  p=  steel  ratio  considered  as  a  thin  cylinder  of  equivalent 
area  surrounding  the  concrete;  As=  cross-section  of  this  steel  cylinder; 
r=  radius.  Then 


A8  =  pxr*     and     thickness  of  cylinder  =       -  =  p—. 

With  no  steel  banding  the  stress  /<•'  would  cause  a  proportionate  lateral 

t  / 
swelling  of  -^-u.     If  the  actual  stress  in  the  steel  is  /8  then  the  compression  per 

&c 

eq.  in.  developed  in  the  concrete  by,  the  steel  reinforcement  =/8p—  .-*-r=^- 

This  compression  caused  by  the  banding  is  equal  in  all  horizontal  directions, 
and  has  the  same  effect  on  distortion  as  two  pairs  of  equal  compressive  forces 
acting  on  two  sets  of  faces  of  a  cube.  The  resultant  lateral  compression  due 

to  these  horizontal  forces  is  equal  to  ^  (1  —  //).     Combining  this  compression 

4Cjc 

with  the  lateral  swelling  caused  by  /</  we  have  the  net  lateral  deformation 
equal  to  '-^IJL—~^-(\  —  H}.  This  net  deformation  must  equal  the  actual 

-&C  ^^C 

deformation  in  the  steel  under  the  stress  f8,  which  is  -§•  or  -^r.     Hence  we 

J^s       nJbc 

have 

£»_Ma-^-A 

Ef     2Ec(l     *      nEc' 

A~part  of  /<•'  may  be  considered  to'  be  balanced  by  the  lateral  compression 
of  ^;    it  is  the  unbalanced  portion  only  which  is  significant.     Call  this 

unbalanced  portion  /c;  then  /c'=/c+i|P.  Then  eliminating  /«,  from  these 
two  equations  we  find  for  /c'  the  value 

.....  w 


132  GENERAL    THEORY.  [Cn.  III. 

At  the  latter  value  eqs.  '!)  and  (2)  would  become  fc'=fe 
(1+ np/16) ,  and  fs  =  \nfc.  Comparing  these  equations  with  those 
of  Art.  95  it  would  appear  that  within  the  limit  of  elasticity  the 
hooped  reinforcement  is  much  less  effective  than  longitudinal 
reinforcement;  in  fact  it  would  seem  that  very  little  stress  can 
be  developed  in  the  steel  under  elastic  conditions  as  here 
assumed.  Such  reinforcement  may,  however,  be  quite  effective 
in  increasing  the  ultimate  strength  of  a  column. 

Results  of  tests  appear  to  accord  in  a  general  way  with 
these  theoretical  relations.  Hooped  columns  have  a  relatively 
large  deformation,  reaching  at  an  early  stage  a  deformation 
equal  to  the  maximum  for  plain  concrete.  Under  further 
loading  the  concrete  is  prevented  by  the  banding  from  actual 
failure,  but  continues  to  compress  and  to  expand  laterally, 
increasing  the  tension  in  the  bands,  the  elasticity  of  the  bands 
rendering  the  column  in  large  degree  still  elastic.  Final  failure 
occurs  upon  the  breakage  of  the  bands  or  their  excessive 
stretching.  Banded  columns  thus  exhibit  a  toughness  or 
ductility  much  greater  than  other  forms,  but  without  a  cor- 
responding increase  hi  stiffness  under  lower  loads.  Ultimate 
failure  is  likely  to  be  long  postponed  after  the  first  signs  of 
rupture,  and  the  column  will  sustain  greatly  increased  loads 
even  after  the  entire  failure  of  the  shell  of  concrete  outside 
the  bands, 

Consid&re  has  made  extensive  theoretical  and  experimental 
investigations  of  hooped  columns,  from  which  he  concludes  that 

We  also  have 


For  ordinary  values  of  p  eqs.  (a)  and  (6)  are  reduced  approximately  to 


and 

/a=/m/c  .......  (2) 


§  966.]  STRENGTH    OF   COLUMNS.  133 

the  ultimate  strength  is  given  by  the  formula 

A,     .....     .     (5) 


in  which  fc  is  the  strength  of  concrete  and  fel  is  the  elastic-limit 
strength  of  the  steel.  This  formula  virtually  counts  the  steel 
worth  2.4  times  as  much  as  in  longitudinal  reinforcement. 
(For  further  discussion  see  Chapter  IV.) 

96a.  Columns  with  both  Longitudinal  and  Hooped  Reinforce- 
ment. —  From  the  theoretical  consideration  of  the  preceding 
article  it  would  appear  that  the  addition  of  bands  or  hoops  to 
columns  having  longitudinal  reinforcement  would  not  have  a 
large  effect  upon  the  deformation  of  such  columns  up  to  the 
point  of  failure  of  the  columns  without  hooping.  The  effect 
of  such  hooping  would  be  rather  to  increase  the  ultimate  possible 
deformation  and,  to  a  less  extent,  the  ultimate  strength.  It 
would  thus  insure  the  integrity  of  the  concrete  up  to  a  deforma- 
tion corresponding  to  the  elastic  limit  of  the  longitudinal 
steel,  but  below  such  limit  it  can  hardly  come  much  into  action. 
Kesults  of  tests  discussed  in  Chapter  IV  bear  out  these  con- 
clusions. 

966.  Long  Columns.  —  For  columns  of  such  a  length  that 
flexural  strength  and  stiffness  become  of  importance  (more 
than  about  20  diameters)  the  working  stress  should  be  reduced 
by  the  use  of  a  long  column  formula.  Until  more  data  are 
available  from  tests,  the  authors  would  propose  the  use  of  the 
theoretical  form  of  Rankine's  formula  which  is,  for  pivoted 
ends, 


in  which  Pf  is  the  strength  of  a  short  column,  and  /  and  E  are 
the  ultimate  strength  and  the  modulus  of  elasticity  of  the 
concrete.  This  formula  gives  results  materially  too  low  when 
applied  to  steel  columns,  but  it  is  believed  that  it  is  not  too 


134  GENERAL    THEORY.  [Cn.  III. 

conservative  for  material  like  concrete.  The  value  of  f/E 
should,  for  conservative  design,  be  taken  at  its  maximum 
rather  than  minimum  value,  say  Viooo,  giving  finally  the 
formula 


1  + 


10,000  \r 


(2) 


For  fixed  ends  the  constant  in  the  denominator  may  be 
made  1/2ooo^  giving 


20,000  \r 

It  may  be  observed  that  if  formulas  be  derived  from  the 
Rankine-Gordon  formulas  for  steel  columns,  by  taking  account 
of  the  difference  in  ultimate  strength  and  modulus  of  elasticity 
of  the  materials,  the  resulting  formulas  would  contain  con- 
stants of  very  nearly  the  same  value  as  for  steel,  namely, 
V  isooo  and  Vseooo-  The  low  values  above  given  represent  a 
larger  degree  of  safety,  which  is  to  be  desired.  For  a  value 
of  I  IT  of  100,  or  a  length  of  about  30  diameters,  the  formula 
for  pivoted  ends  gives  an  ultimate  strength  of  two-thirds  that 
of  the  short  column.  Because  of  the  fact  that  it  is  difficult 
to  secure  thoroughly  homogeneous  concrete,  and  that  variations 
in  quality  will  affect  the  strength  of  long  columns  more  seriously 
than  any  other  structural  form  long  columns  should  generally 
be  avoided, 


CHAPTER  IV. 
TESTS  OF  BEAMS   AND  COLUMNS. 

BEAMS. 

97.  Methods  of  Failure  of  a  Reniforced-concrete  Beam. — 

A  reinforced-concrete  beam  tested  to  destruction  will  usually 
fail  in  one  of  three  ways: 

(a)  By  the  jielding  of  the  steel  at  or  near  the  section 
of  maximum  bending  moment. 

(6)  By  the  crushing  of  the  concrete  at  the  same  place. 

(c)  By  a  diagonal  tension  failure  of  the  concrete  at  a 

place  where  the  shear  is  large. 

Methods  (a)  and  (b)  may  be  called  " moment"  failures.  Method 
(c)  is  sometimes  called  a  shear  failure,  but  this  term  is  some- 
what misleading,  as  the  concrete  in  such  cases  does  not  fail  by 
shearing. 

(a)  As  a  beam  is  progressively  loaded  and  the  steel  has 
reached  its  yield  point  any  further  load  will  rapidly  increase 
the  deformation.  The  effect  of  this  is  to  open  up  large  cracks 
in  the  tension  side  and  to  raise  the  neutral  axis.  This  causes 
a  rapid  increase  in  the  compressive  stress  in  the  concrete  and 
ultimate  failure  soon  occurs  by  the  concrete  crushing.^  Such 
yielding  may  also  result  in  final  failure  by  diagonal  tension 
if  large  shear  exists  near  the  place  of  maximum  moment.  In 
this  case  the  primary  cause  of  failure  is  the  yielding  of  the  steel 
and  such  failure  may  properly  be  called  a  tension  failure.  The 
additional  load  carried  after  the  yield  point  is  reached  depends 
on  the  excess  strength  of  the  concrete,  position  of  loads,  and 

135 


136  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV. 

other  causes,  but  it  is  usually  not  large  and  cannot  be  safely 
considered.  The  yield  point  of  the  steel  may  therefore  be  con- 
sidered its  ultimate  strength  for  reinforcing  purposes. 

(6)  If  the  beam  is  relatively  long  and  the  amount  of  steel 
is  sufficient  so  that  the  crushing  strength  of  the  concrete  is 
reached  before  the  yield  point  of  the  steel,  a  failure  by  crushing 
is  likely  to  result.  In  this  case  tension  cracks  may  appear,  but 
will  not  become  large.  Fig.  38,  (a)  and  (6),  illustrates  methods 
of  failure  (a)  and  (6)  respectively. 


/  I 

/I,,. 

• 

1 

(d) 

t 

(&) 


FIG.  38.— Methods  of  Failure  of  Beams. 

(c)  Diagonal  tension  failures  are  likely  to  occur  whenever 
large  shearing  stresses  exist  together  with  considerable  hori- 
zontal or  moment  stresses,  and  when  no  special  provision  is 
made  for  such  conditions.  This  is  especially  likely  to  occur 
in  beams  of  relatively  great  depth,  beams  having  a  ratio  of 
depth  to  length  of  more  than  about  1:10  being  likely  to 'fail 
in  this  way  if  no  special  provision  is  made  for  web  rein- 
forcement.y 

Fig.  38,  (c),  illustrates  the  typical  diagonal  tension  failure 
where  only  horizontal  bars  are  used.  The  initial  crack  forms 
at  a.  This  gradually  extends  upwards  in  an  inclined  line  and 


1A&4 


§99.]  METHODS  OF  FAILURE  OF  BEAMS.  141 

a  little  later  the  concrete  begins  to  fail  in  a  horizontal  tension 
crack  just  above  the  rods,  progressing  from  a  towards  the  end 
of  the  beam.  Tension  along  this  line  is  brought  about  by  the 
new  conditions  existing  after  the  concrete  has  become  cracked 
along  the  diagonal  line  and  the  normal  diagonal  tension  has 
thus  ceased  to  act.  Usually  this  horizontal  crack  rapidly  ex- 
tends .to  the  end  of  the  beam  and  the  failure  is  complete.  In 
other  cases  the  diagonal  crack  may  extend  to  the  top  of  the 
beam,  allowing  the  part  on  the  right  to  drop  down  and  causing 
final  failure.  In  such  a  case  the  concrete  on  the  left  may  remain 
intact.  Figs.  39  and  40  are  photographs  representing  " diagonal- 
tension  "  failures. 

A  rupture  of  the  concrete  on  a  diagonal  line  also  causes 
an  increase  in  the  stress  on  the  rod  at  a,  as  shown  more  fully  in 
Art.  108.  This  may  result  in  a  failure  of  bond,  especially  if 
the  support  is  too  near  the  end  of  the  beam. 

Final  failure  thus  often  results  from  stresses  which  are  devel- 
oped after  initial  failure  has  occurred,  and  while  the  cause 
of  final  failure  is  important  from  the  standpoint  of  ultimate 
strength,  yet  of  more  importance  in  design  is  the  initial  failure 
and  its  cause.  Other  conditions  besides  those  already  men- 
tioned may  influence  final  failure  so  as  often  to  mislead  the 
observer  as  to  the  cause  of  the  initial  failure. 

98.  Minor  Causes  of  Failure.-f-S\ippmg  of  the  bars  may 
cause  failure,  but  under  usual  conditions  it  will  not  occur  ; 
and  as  it  can  readily  be  obviated  by  proper  construction)  it  need' 
not  be  considered  as  limiting  the  strength  of  the  beam.  (Failure 
by  the  shearing  of  the  concrete  near  the  support  is  possible  where 
the  load  is  very  close  thereto,  but  as  the  shearing  strength  of 
concrete  is  about  one-half  the  crushing  strength,  such  failures 
are  exceedingly  unlikely)and  need  rarely  be  considered.  The 
usual  so-called  " shear"  failures  are  in  reality  diagonal-tension 
failures. 

99-  Tests  of  Beams  Giving  Steel-tension  Failures. — The 
diagrams  of  Figs.  41  and  42  present  in  a  roughly  classified  form 
results  of  the  most  important  tests  on  reinf orced-concrete  beams 


142  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV. 

in  which  the  failure  appears  to  have  been  caused  primarily  by 
the  yielding  of  the  steel.  In  such  a  case  the  strength  of  the 
beam  is  directly  proportional  to  the  elastic-limit  strength  of 
the  steel,  and  hence  the  tests  have  been  classified  as  nearly  as 
practicable  with  respect  to  this  limit.  The  tests  are  thus 
divided  into  four  groups  according  tc  values  for  the  elastic 
limit  as  given  in  the  diagrams.  On  each  of  the  diagrams  are 
drawn  theoretical  curves  of  strength  using  values  for  the  steel 
stress  corresponding  to  the  elastic  limit  for  the  group.  The 
full  line  is  based  upon  the  parabolic  law  of  stress  variation,  the 
full  parabola  being  used;  the  dotted  line  is  based  upon  the 
straight-line  law  of  stress  variation.  The  value  of  n  was  taken 
at  15.* 

Considering  the  nature  of  the  material  and  of  the  tests  the 
agreement  between  theory  and  experimental  results  is  very  satis- 
factory. It  is  to  be  expected  that  the  theoretical  values 
should  represent  minimum  rather  than  average  results,  since 
the  strength  of  a  beam  as  determined  by  the  elastic  limit  of  the 
steel  should  be  at  least  equaled,  and  generally  slightly  exceeded, 
in  a  test  if  failure  does  not  occur  in  some  other  way.  If  the 
conditions  are  favorable  the  strength  may  considerably  exceed 
that  corresponding  to  the  elastic  limit  of  the  steel,  and  in  a 
few  tests  the  steel  has  been  pulled  apart  before  complete  col- 
lapse has  taken  place.  Such  excess  of  strength  cannot  be 
counted  upon,  however,  as  is  well  indicated  in  the  diagrams. 


*  The  sources  of  information  are  as  follows: 

1.  Boston  Transit  Commission,  Fourth  Annual  Report,  1904. 

2.  Bulletins  Nos.  1  and  4,  University  of  Illinois,  Engineering  Ex- 

periment Station 

3.  Jour.  West.  Soc.  Eng.,  vol.  X,  1905,  p.  705  (C.,  M.  &  St.  P.  R'y 

Co.'s  tests). 

4.  Jour.  West,  Soc.  Eng.,  Vol.  IX,  1904,  p.  239  (tests  of  M.  A.  Howe). 

5.  Bulletins  No.   4,  Vol.   3,  and  No.  1,  Vol.  4,  Engineering  Series, 

University  of  Wisconsin,  1907. 

6.  Proc.  Am.  Soc.  Test.  Materials,  Vol.  IV,  1904,  p.  508  (Univ.  of 

Penn.  tests). 

7.  Eng.  Record,  Vol.  LI,  1905,  p.  545  (Purdue  Univ.  tests). 


TESTS  OF  BEAMS. 


143 


% 


•  Boston  Transit  Commission 
O  University  of  Illinois 
0  C.  M.  &  St.  P.  Ry. 
x  University  of  Wisconsin 
8  M.  A.  Howe 


0.5  1.0  1.5 

Percentage  Beinfocc.ement 

FIG.  41. — Steel-tenison  Failures. 


2.0 


2-5 


144 


TESTS  OF  BEAMS  AND  COLUMNS. 


[On.  IV. 


1400 
1300 
1-200 
1100 
1000 
900 
800 
700 
600 
500 
5^400 

V 

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100 
0 
700 
600 
600 
400 
300 

200 
100 

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•  Boston  Transit  Commission 
o  University  of  Illinois 
0  C.  M.  &  St.  P.  Ry. 
Q  University  of  Pennsylvania 
A  Purdue  University 

f 

ft 

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FIG.  42. — Steel-tension  Failures. 


§  100.]  TESTS  OF  BEAMS.  145 

In  the  tests  of  the  Boston  Transit  Commission,  which  range 
uniformly  high,  the  conditions  were  favorable,  inasmuch  as 
the  beams  were  tested  with  center  load.  The  concrete  was 
also  of  very  high  grade,  having  a  crushing  strength  of  about 
4000  lbs/in2,  thus  enabling  the  steel  to  elongate  very  con- 
siderably before  final  failure  occurred  through  the  crushing 
of  the  concrete. 

No  distinction  has  been  made  in  these  diagrams  between  the 
different  grades  of  concrete  employed.  Variations  in  concrete 
will  affect  the  results  only  by  slightly  affecting  the  position 
of  the  neutral  axis,  and  hence  the  resisting  moment  of  the 
steel,  and  by  postponing  somewhat  the  final  failure,  as  noted 
above. 

Later  tests  by  Talb'ot  *  gave  tension  failures  in  which  the 
calculated  steel  stress  closely  agreed  with  the  yield-point  of  the 
metal.  His  tests  also  showed  a  somewhat  higher  strength  for 
a  center  load  than  for  loads  at  two  or  more  points. 

100.  Results  from  Individual  Tests. — Numerous  tests  of 
beams  have  been  made  in  which  extensometers  have  been  used 
to  measure  distortions  so  that  the  deformation  of  the  steel 
and  of  the  extreme  fiber  of  the  concrete  could  be  calculated 
and  the  neutral  axis  determined.  Results  of  such  measure- 
ments of  deformations  and  also  of  center  deflections  are  shown 
in  Figs.  43  and  44  for  two  typical  beams.  In  Fig.  43  the 
proportions  were  such  that  the  failure  occurred  by  diagonal 
tension;  neither  the  steel  nor  the  concrete  was  stressed  to  the 
limit  of  failure.  During  the  first  stage  of  the  test,  up  to  a  load 
of  about  2500  pounds,  the  deformations  in  both  steel  and  concrete 
are  proportional  to  the  loads.  Up  to  this  point  the  tension 
deformation  has  not  been  great  enough  to  begin  to  rupture 
the  concrete,  but  with  increasing  loads  arid  deformations 
the  concrete  begins  to  fail,  as  shown  by  the  appearance  of 
minute  cracks  (the  "water-marks"  discussed  in  Art.  42),  indi- 
cated on  the  diagram  by  the  letters  W  M.  The  deformation 

*  Bull.  No.  14,  Univ.  of  111. 


146 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn,  IV, 


Load  in  Pounds  _. 

no 

,/ 

7 

^ 

x 

^ 

y/ 

^ 

X 

Ow 

s 

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.3     014     0 

in  Inches 
5     016      017     0 

8     0 

9      1 

0 

0              .0004           .0008          .0013           .0016            .0030           .0034            .0038 
Deformation  per  Unit  Length 

FIG.  43. 

Deflection  in  Inches 

Ol3     04     0.5      06     0.7     08      09      10 


.0008  .0012      -    .0016 

Deformation  per  Unit  Length 

FIG.  44. 


.0028 


§  101.]  TESTS    OF  BEAMS  147 

at  the  first  "  water-mark"  in  this  case  was  about  .00018,  corre- 
sponding to  a  stress  of  270  lbs/in2,  assuming  a  modulus  of 
elasticity  of  1,500,000.  The  first  visible  crack  appeared  at  the 
point  marked  C. 

The  failure  of  the  concrete  in  tension  takes  place  somewhat 
gradually  and  causes  a  gradual  increase  in  the  rate  of  deforma- 
tion as  indicated  by  the  curved  part  c/f  the  diagram  between 
loads  of  2500  and  4000  pounds.  After  the  concrete  has  ceased 
to  offer  any  considerable  resistance  in  tension  the  deformations 
again  become  nearly  proportional  to  the  loads,  but  at  a  different 
ratio  from  that  obtaining  previously,  giving  nearly  straight  lines 
for  both  steel  and  concrete— in  this  case  to  the  end  of  the  test. 

In  Fig.  44  the  amount  of  steel  was  small  and  a  tension 
failure  occurred.  This  is  indicated  by  the  great  deformations 
at  the  end  of  the  test.  The  curves  in  the  early  stages  of  the 
test  are  very  similar,  in  general  form,  to  those  in  Fig.  43. 

In  the  case  of  a  compressive  failure  the  curve  for  compres- 
sion shows  an  increased  rate  of  deformation  towards  the  end, 
somewhat  similar  to  the  diagram  for  simple  compression. 

10 1.  Position  of  Neutral  Axis  and  Value  of  n—  Reference  to 
the/analysis)  of  Arts.  55  and  5^'  show  that  in  the  calculation  of 
the  strength  of  reinforced  beams  the  determination  of  the 
position  of  the  neutral  axis  is  of  prime  importance.  This 
being  known,  the  strength  can  be  determined  with  little  uncer- 
tainty. In  determining  the  position  of  the  neutral  axis 
eq.  (1)  of  Art.  55  shows  it  to  depend  only  upon  the  amount  of 
steel  used  and  upon  the  ratio  Es/Ec  or  n.  The  only  element 
of  uncertainty  is  the  value  of  EC.  This  is  the  modulus  of  elas- 
ticity of  the  concrete  in  compression  and  it  might  be  considered 
sufficient  to  take  the  value  as  determined  in  the  ordinary 
compression  test.  However,  the  variation  of  Ec  for  different 
stresses,  and  the  effect  of  the  tensile  stresses  in  the  concrete 
below  the  neutral  axis  (a  stress  which  is  properly  not  allowed 
for  in  the  resisting  moment),  make  it  desirable  to  compare 
experimental  determinations  of  the  neutral  axis  with  theo- 
retical position  for  various  assumed  values  of  EC  or  of 


148 


TESTS    OF    BEAMS    AND    COLUMNS. 


[Cn.  IV. 


Many  experiments  have  been  made  in  which  the  position  of 
the  neutral  axis  has  been  determined.  Among  the  best  are 
those  by  Bach,*  made  on  1:4  gravel  concrete,  6  to  7  months 
old.  The  beams  were  2  m.  long  and  30  cm.  deep  and  were 
loaded  at  quarter  points.  The  observed  positions  of  the  neutral 
axis  (values  of  &),  at  various  loads  are  given  in  the  following 
table: 

POSITION  OF  NEUTRAL  AXIS. 

(VALUES  OF  fc.)     (BACH.) 


No.  of 

Percent 

Values  of  k  for  Various  Proportions  of 
Ultimate  Load. 

Theoretical  Values. 

Beams. 

Reinf. 

Initial 

i  Load. 

iLoad. 

|  Load. 

Full  Load 

for  n  =  12 

for  n=15 

5 

0.54 

.56 

.53 

.43 

.33 

.31 

.30 

.33 

3 

0.43 

.59 

.55 

.47 

.31 

.28 

.27 

.30 

5 

1.32 

.59 

.55    ' 

.45 

.44 

.46 

.43 

.46 

"(  The  theoretical  positions  are  also  given  for  ft =12  and 
n= 15.  The  value  of  E  for  this  concrete,  at  a  load  of  600  lbs/in2, 
as  determined  by  compression  tests,  was  3,300,000  lbs/in2. 

Similar  tests  have  been  made  on  T-beams  by  Bach  and 
also  by  Withey.t  All  of  Bach's  tests  and  those  on  T-beams 
by  Withey  are  plotted  in  Fig.  44a. 

In  Figs.  45  and  46  are  plotted  in  a  different  form  results 
of  various  tests  on  rectangular  beams.  On  the  diagrams  are 
also  plotted  the  theoretical  positions  of  the  neutral  axis  for 
various  values  of  n.  The  full  lines  are  based  on  the  straight- 
line  stress  variation  assumption,  and  the  dotted  lines  on  the 
assumption  of  a  parabolic  law  in  accordance  with  Professor 
Talbot's  method  (see  Art.  65).  The  dotted  lines  have  been 
drawn  only  for  a  single  value  of  15  for  n.  For  a  three-quarter 
load  the  dotted  line  for  n  =  15  would  coincide  very  closely  with 
the  full  line  for  ft  =  20.  The  value  of  q  has  been  taken  at  i, 
J,  and  J,  respectively. 


*  Mit.  iiber  Forsch.  a.  d.  Gebiet  des  Ing..  1907,  45-47. 
t  Bull.  Univ.  of  Wis.,  1908,  Vol.  4,  No.  2. 


101.] 


TESTS    OF    BEAMS. 


149 


It  will  be  noted  that  for  the  lower  loads  and  the  small 
percentages  of  steel  the  neutral  axis  is  more  uncertain  and 
generally  lower  than  for  the  higher  loads  and  larger  percen- 
tages. This  is  due  to  the  relatively  large  influence  of  the 
tensile  strength  of  the  concrete  in  such  cases.  The  T-beam 

Proportionate  Load 


ja  Values  of  k 
o  bo  bs  "*.  *o  o 

o      x      y*      &     i  o     K      K      %     i  o     K      > 

i           K           1 

.2 
.4 

.6 

.8 

1.0 
0 

.2 
.4 

.6 
.8 
1  0 

w*=12 

n=l2 

^ 

^ 

' 



^ 

n=is 

^^ 

•  ! 

===: 

p- 

0.43^ 

t- 

0.54^ 

r- 

u* 

Bectai 

gular 

Jeiinis 

rBach) 

.   ...  _  _.  „.. 

n=l2 

^*~  • 

* 



n-12 

._—  —  ••* 

n—  i5 

^" 

n=!5 

t- 

u* 

p  = 

W}6 

T—  beams  (Bach) 
FIG.  44a. — Position  of  Neutral  Axis. 


10       K        X 

T-beams/Wlthey) 


tests  show  relatively  little  of  this  effect  owing  to  the  small 
area  of  concrete  in  tension. 

From  these  results  it  appears  that  a  value  of  15  for  n  is 
not  too  large  for  calculations  of  strength  of  beams  under  the 
usual  assumptions.  This  value  is  the  one  most  generally 
used,  but  a  value  of  12  is  also  frequently  employed.  The 
value  of  15  corresponds  to  a  value  of  Ec  of  2,000,000,  which 
is  somewhat  low  as  determined  by  compressive  tests. 
A  value  of  n=10,  corresponding  to  #=3,000,000,  does  not 


150 


TESTS  OF  BEAMS  AND  COLUMNS. 


[CH.  IV. 


At  One-fourth  Ultimate  Load 
0.5  1.0  1.5  2.0 


0.2 
0.4 

0.6 

0.8 

1.0 

0 

,0.2 
!0.4 


:§0.6 


At  One-half  Ultimate  Load 


At  Three-fourths  Ultimate  Load 


0,5 


1.0  1.5 

Percentage  Reinforcement 


Fio.  45.— Position  of  Neutral  Axis.    (1: 2:  5  Concrete.) 


§ ioij- 


POSITION  OF  NEUTRAL  AXIS. 


151 


At  One-fourth  Ultimate  Load 


At  One-half  Ultimate  Load 


At  Three-fourths  Ultimate  Load 


1.0  1.5  2.0 

Percentage  Reinforcement 

FIG.  46.— Position  of  Neutral  Axis.     (1:3:6  Concrete.) 


152  TESTS    OF    BEAMS    AND    COLUMNS.  [Cn.  IV. 

accord  with  results  from  bending  tests.  If  the  comparison 
between  measured  and  calculated  positions  of  the  neutral  axis 
be  made  on  the  basis  of  the  parabolic  law  of  stress  variation  the 
results  will  differ  considerably  in  the  latter  stages  of  the  tests, 
but  very  slightly  at  the  quarter  load.  It  should  be  noted, 
however,  that  it  is  only  with  the  high  percentages  of  steel  that 
the  concrete  stress  reaches  nearly  to  its  ultimate  value,  and 
hence  is  the  only  condition  where  the  full  parabolic  law  can  be 
expected  to  give  consistent  and  rational  results. 

In  some  of  the  tests  whose  results  are  plotted  here  the  con- 
crete was  cut  away  from  the  steel  for  the  measured  distance, 
leaving  it  exposed.  The  position  of  the  neutral  axis  was  very 
slightly  affected. 

1 02.  Observed  and  Calculated  Stresses  in  Steel. — Where 
the  neutral  axis  is  determined  by  extensometer  measurements 
a  check  upon  theoretical  results  can  be  obtained  by  calculating 
the  stress  in  the  steel  in  two  ways:  (1)  from  the  observed 
deformations  at  the  plane  of  the  steel,  and  (2)  from  the  known 
bending  moment  and  known  position  of  the  neutral  axis.  In 
the  first  calculation  the  tensile  strength  of  the  concrete,  which 
is  neglected,  causes  some  error,  especially  under  light  loads, 
and  in  the  second  calculation  the  exact  position  of  the  centroid 
of  pressure  in  the  concrete,  especially  in  the  later  stages  of  the 
test,  is  to  a  small  degree  uncertain,  but  as  the  variation  in 
steel  stress  is  only  about  2%,  using  the  two  extreme  assump- 
tions of  stress  variation,  this  source  of  error  is  not  great.)  Table 
No.  7  presents  several  representative  results  derived  from  such 
calculations.  The  stresses  calculated  from  moments  are  based 
on  the  assumption  that  the  concrete  takes  no  tension.* 

Tests  have  been  made  at  the  University  of  Illinois  and  at 
the  University  of  Wisconsin  in  which  the  rods  have  been  exposed 
for  a  considerable  distance  along  the  center  of  the  beam,  and 
thus  have  been  much  less  affected  by  any  possible  tensile 
stress  in  the  concrete.  Measurements  of  extension  made  in  such 

*  For  further  data  see  Table  No.  12,  Art.  112. 


f  103.1 


POSITION  OF  NEUTRAL  AXIS. 


153 


cases  show  little  variation  from  those  made  on  the  ordinary 
beam. 

TABLE  No.  7. 

STRESSES  IN  STEEL  REINFORCEMENT. 


Calculated  Stress  in  SteeL 

Observed 

lbs/in2. 

Authority 

Per  Cent 
Retnf  or  cement  . 

Position  of 
Neutral  Axis, 

k. 

From 

From  Exten- 

Moments. 

sions  in  Steel. 

.74 

.410 

33,100 

36,000 

Talbot; 
Bull.    Univ.    of 
III,  1906. 

1.23 
1.60 
1.66 
1.84 

.470 
.501 
.505 
.606 

35,000 
29,500 
30,600 
25,600 

36,000 
35,400 
30,000 
27,200 

1.84 

.552 

28,300 

30,000 

Withey; 
Bull.    Univ.    of 
Wis.,  1907. 

2.9 
2.9 

.670 
.fr  0 

35,200 
31,600 

36,000 
33,000 

Considering  the  nature  of  such  experiments  the  results 
obtained  may  be  considered  as  according  with  theory  very  sat- 
isfactorily. 

103.  Compressive  Stresses  in  Concrete  in  Beams  and 
in  Compression  Specimens. — An  important  question  relating 
to  proper  working  stresses  is  whether  the  ultimate  compressive 
strength  of  concrete  in  a  beam  is  the  same  as  determined  by  a 
direct  compression  test. 

The  results  of  certain  tests  indicate  that  the  compressive 
strength  and  ultimate  deformation  in  a  beam  may  be  some- 
what greater  than  in  a  prismatic  compressive  piece;  and  ii 
would  seem  that  the  differences  in  condition  are  sufficient  to 
make  such  a  difference  possible.  In  a  compressive  specimen 
the  material  is  free  to  shear  in  any  direction,  thus  limiting 
the  strength  of  the  specimen  to  its  weakest  shearing  plane.  In 
a  beam  the  (shear)  failure  is  practically  confined  to  planes 
perpendicular  to  the  side  of  the  beam.  Furthermore,  in  a 
beam  the  material  is  not  subjected  to  the  secondary  stresses 
due  to  possible  poor  bedding  of  the  test  specimen  or  non- 


154 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 


parallel  motion  of  the  testing  machine,  as  is  the  case  in  compres- 
sion tests. 

In  most  of  the  tests  reported  both  the  beams  and  the  ac- 
companying compression  specimens  have  been  hardened  in 
air.  Under  these  conditions  there  is  usually  some  drying-out 
effect  resulting  in  a  weaker  concrete  than  if  hardened  in  water, 
and  owing  to  the  smaller  dimensions  of  the  compressive  speci- 
mens the  effect  will  be  greater  with  them  than  with  the  rela- 
tively large  beams.  Many  tests  have  therefore  shown  a  com- 
pressive strength  of  concrete  in  the  beam  considerably  greater 
than  results  obtained  on  cubes.  When  both  beam  and  cube 
are  hardened  in  water  the  results  do  not  differ  greatly.  The 
following  are  some  results  obtained  on  tests  made  relative 
to  this  point.*  The  beams  were  51"X6"  net  section  and 
5  ft.  span.  They  were  reinforced  with  2^%  of  steel  and 
gave  compressive  failures.  The  cubes  were  4  inches  in  dimen- 
sion and  the  cylinders  6"  in  diameter  by  18"  high. 


Stress  in  Concrete  at  Rupture,  lbs/in2. 

Beam. 

Cube. 

Cylinder. 

Hardened 
Hardened 

f  1 

1770 
1460 

1810 
1850 

1187 
1350 

1450 
1750 

1380 
1295 

1265 
1680 

in  air        <2 

(3   . 

in    water  <  .    . 

The.  stresses  in  the  beams  were  calculated  on  the  basis  of  the 
parabolic  variation  of  stress,  the  neutral  axis  being  determined 
by  extensometers. 

It  will  be  seen  that  in  case  of  the  specimens  hardened  in 
air  there  is  a  marked  difference  in  strength,  but  where  hardened 
in  water  the  difference  is  much  less.  The  difference  is  hardly 
sufficient  to  warrant  much  consideration  in  the  determination 
of  working  stresses. f 

*  Bulletin  No.  1,  Vol.  4,  Engineering  Series,  University  of  Wisconsin,  1907. 
t  For  further  data  see  Table  No,  12,  Art.  112. 


§  105.]  POSITION   OF  NEUTRAL  AXIS.  155 

104.  Conclusions    Regarding    Moment    Calculations. — 

(The  comparison  of  experimental  results  with  theoretical  analysis 
herein  given  shows  that  the  simple  beam  theory  as  generally 
employed,  neglecting  the  tension  in  the  concrete,  can  be  used 
with  confidence.  In  particular,  the  results  appear  to  show 
that  calculated  on  the  basis  of  such  theory  the  yield  point 
(commonly  called  the  elastic  limit)  of  the  steel  may  safely  be 
taken  as  its  ultimate  strength  in  reinforced  beams;  that  the 
crushing  strength  of  concrete  as  determined  by  tests  on  cubes 
hardened  under  exactly  similar  conditions  as  the  beams  will  be 
fully  realized  in  the  beam;  that  for  working  loads  the  straight- 
line  law  of  stress  variation  is  sufficiently  exact;  that  the  value 
of  n  may  be  taken  at  about  15,  but  that  great  accuracy  in  this 
respect  is  unnecessary;  that  for  ultimate  values,  especially 
where  the  concrete  is  near  failure,  the  parabolic  assumption 
of  stress  variation  may  well  be  used. 

105.  Tests  in  which  Failure  Occurred  by  Diagonal  Ten- 
sion.   Influences  Affecting  Failure  by  Diagonal  Tension. — The 
strength  of  a  beam  in  diagonal  tension  is  not  a  simple  function 
of  the  shear,  but  as  shown  in  Art.  90  it  depends  also  upon  the 
horizontal  tension  or  bending-moment  stresses  in  the  concrete. 
These  will  in  turn  depend  upon  the  actual  bending  moment 
at  the  section  of  failure  and  the  amount  of  horizontal  reinforce- 
ment a  large  percentage  of  reinforcement  reducing  the  horizontal 
deformation  and  therefore  the  tension  in  the  concrete  and 
tending  to  strengthen  the  beam  as  regards  failure  in  diagonal 
tension.    The  strength  of  the  beam  therefore  depends  upon  the 
relation  between  shear  and  bending  moment  and  upon  the 
amount  of  reinforcement.     The   chief   factor  is,  however,  the 
shearing  stress. 

From  the  preceding  considerations  it  is  evident  that  the 
nature  of  the  loading  will  influence  the  strength  of  the  beam. 
Most  structures  are  calculated  for  uniform  or  approximately 
uniform  loading,  and  in  experimental  work  two  concentrated 
loads  applied  at  the  third  points  are  commonly  used  as  repre- 
senting roughly  the  conditions  which  exist  in  the  uniformly 


156 


TESTS  OF  BEAMS  AND   COLUMNS. 


[Cn.  IV. 


loaded  beam.  Fig.  47  represents  the  variation  in  moment 
and  shear  in  a  beam  loaded  at  the  third  points,  while  Fig.  48 
shows  similar  curves  for  a  uniformly  loaded  beam.  It  is 
to  be  noted  that  in  the  first  case  maximum  shear  occurs  where 
maximum  moment  exists,  while  in  the  latter  case  maximum 
shear  occurs  at  the  point  of  zero  moment.  In  the  former  case 


Shear 


FIG.  47.  FIG.  48. 

diagonal-tension  failure  will  occur  just  outside  the  loads,  while 
in  the  latter  case  it  will  occur  nearer  to  the  support  where  the 
moment  is  considerably  less  than  the  maximum.  Conditions 
as  to  shear  are  therefore  somewhat  more  favorable  in  the  con- 
tinuously loaded  beam.  A  single  concentrated  load  causes  less 
shear  for  a  given  moment  than  the  double  load,  and  is  there- 
fore more  favorable  as  regards  shear. 

As  continuous  beams  are  commonly  used  in  building  con- 
struction it  will  be  useful  to  note  here  the  variation  in  shear 
and  moment  in  such  a  beam.  This  is  shown  in  Fig.  49,  and  it 
will  be  seen  that  the  conditions  here  are  quite  unfavorable, 
large  shear  occurring  near  the  supports  where  the  negative 
bending  moment  is  large. 

Whether  a  beam  will  fail  from  moment  stresses  or  shearing 
stresses  will  depend  largely  upon  its  relative  length  and  depth. 
For  any  given  distribution  of  loads  and  given  stresses  there  is 
a  definite  ratio  of  length  to  depth  for  equal  strength  as  given 


§  106.]  WEB  REINFORCEMENT.  157 

in  Chap.  Ill,  Art.  91,  but  by  reason  of  the  variation  in  shearing 
strength  due  to  the  direct  effect  of  moment  and  amount  of 
steel,  these  formulas  can  be  considered  as  only  roughly  approxi- 
mate. 

1 06.  Methods  of  Web  Reinforcement. — There  are  in  use 
many  methods  of  placing  steel  in  the  web  so  as  to  reinforce  it 
against  inclined  tension  failure.  The  various  methods  may,  for 
convenience,  be  divided  into  three  groups:  (1)  Reinforcing 
metal  placed  at  an  inclination;  (2)  Reinforcing  metal  placed 
vertically;  (3)  Miscellaneous  methods. 


Moment 


Shear 
Fio.  49. 

(1)  Theoretically  the  most  effective  way  to  reinforce  against 
tension  failure  in  any  direction  is  to  place  reinforcement  across 
the  lines  of  rupture,  or  in  the  direction  of  the  maximum  tensile 
stresses.  In  the  case  of  web  tension  the  lines  of  maximum 
stress  vary  in  direction,  but  it  is  not  practicable  or  necessary 
to  have  the  inclination  of  the  reinforcing  rods  exactly  the  same 
as  the  lines  of  maximum  tension,  and  various  arrangements 
will  serve  to  accomplish  the  purpose.  The  most  common 
method  is  to  use  several  rods  for  the  horizontal  reinforcement 
and  then  to  bend  a  part  of  these  upwards  as  they  approach  the 
end,  where  they  are  not  needed  to  resist  bending  stresses.  Such 
an  arrangement  is  shown  in  Fig.  50,  (a)  and  (b).  Separate 


158  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV. 

inclined  rods  may  also  be  used,  attached  or  not  to  the  horizontal 
bars.  The  " stirrups"  commonly  placed  in  a  vertical  position 
may  thus  be  inclined.  Special  forms  of  bars  may  be  used,  as 
the  Kahn  bar,  Fig.  7,  p.  31,  in  which  strips  are  sheared  from 
the  main  bar  and  bent  up. 

(2)  Vertical  reinforcement   has  long  been  the    established 
practice  in  European  work  where  the  experience  has  extended 
over  many  years.     It  has  proven  its  effectiveness  and  in  con- 
nection with  bent  rods  has  many  practical  advantages.     Vertical 
reinforcement  usually  consists  of  some  form  of  bent  rod  or 
band  styled  a  " stirrup",  placed  as  shown  in  Fig.  50,  (c)  and 
(d).    The  Hennebique  system,  widely  and  successfully  used, 
employs  both  the  inclined  rods  and  the  vertical  stirrup  (see 
Fig.  85,  Art.  162).     Combined  with  bent  rods  many  arrange- 
ments of  stirrups  are  possible,  especially  in  continuous-girder 
constructions,  the  chief  object  being  to  secure  good  connection 
of  stirrup  to  top  and  bottom  steel. 

(3)  Some  form  of  web  of  woven  wire  or  expanded  metal 
may  be  used  for  web  reinforcement,  and  still  other  arrange- 
ments of  wire  or  rods  employed  as  illustrated  in  Fig.  50,  (e), 
(/),  and  (g).    In  (g),  representing  the  Visintini  system,  the  beam 
is  made  into  a  truss  in  which  the  chords  and  the  tension  diagonals 
are  reinforced. 

107.  Action  of  Web  Reinforcement. — To  aid  in  appreciating 
the  action  of  steel  placed  in  various  ways,  consider  the  typical 
diagonal  tension  failure,  Fig.  51,  as  it  occurs  where  only  hori- 
zontal rods  are  used.  The  inclined  crack  at  a  usually  appears 
first,  due  to  rupture  of  the  concrete  in  tension.  To  assist  in 
preventing  this  rupture  in  its  initial  stage  the  most  efficient 
reinforcement  would  be  such  as  supplied  by  the  inclined  rod  1, 
fastened  to  or  looped  about  the  horizontal  bar,  or  by  the  bent 
end  of  one  of  the  horizontal  bars.  Reinforcement  in  this  direc- 
tion is  in  a  position  to  take  stress  immediately.  The  vertical 
rod  2  can  hardly  be  as  effective  as  the  inclined  rod  in  preventing 
initial  rupture,  for  so  long  as  the  concrete  is  intact  the  deforma- 
tion on  a  vertical  line  is  practically  zero,  owing  to  the  combined 


107.] 


WEB   REINFORCEMENT. 


159 


(a) 


(6) 


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(c) 


(d) 


(e) 


PIG.  50. — Some  Methods  of  Web  Reinforcement. 


160 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 


action  of  web  tension  and  web  compression  at  right  angles  to 
each  other.  Unless  the  unit  stresses  in  the  steel  be  made  very 
low,  however,  it  is  likely  that  the  concrete  has  received  excessive 
tensile  stress  even  under  working  conditions,  and  may  be  assumed 
to  be  ruptured  more  or  less,  in  the  same  manner  as  on  the  tension 
face  of  the  beam  at  points  of  maximum  moment.  At  least  the 
distortion  in  tension  will  be  greater  than  in  compression,  and 
there  will  be  a  vertical  movement  of  the  concrete  on  the  left 
of  the  crack,  a,  downwards  with  respect  to  the  part  of  the  right, 
and  the  vertical  rod  2  will  be  brought  into  direct  action  if 
looped  around  or  attached  to  the  horizontal  bars.  Such  a 
rod  may  then  be  more  effective  (allow  of  less  vertical  movement) 
than  the  inclined  rod.  Practically,  there  is  no  great  difference 
in  the  effectiveness  of  the  two  forms  of  reinforcement  if  closely 
spaced  so  as  to  be  in  position  to  prevent  excessive  deformation 
all  along  the  lower  portion  of  the  beam.  To  secure  thoroughly 
effective  reinforcement  in  this  respect  requires  very  careful 


FIG   50«. 

arrangement  of  the  rods  and  faithful  execution  of  the  work. 
The  action  of  vertical  stirrups  in  resisting  vertical  deforma- 
tion and  the  extension  of  inclined  cracks  is  very  well  illus- 
trated in  Fig.  50a,  which  represents  the  conditions  which 
developed  in  a  test  of  a  beam.  The  cracks  are  numbered 
in  the  order  of  their  appearance,  final  failure  occurring  at 
crack  No.  4  and  being  due  to  inadequate  web  reinforcement. 
The  stirrups  were  stressed  beyond  their  yield-point. 

Vertical    stirrups  spaced   a    distance    apart    equal   to    or 
greater  than    the    depth  of   the  beam   will  give  little  aid  in 


§  109.]  WEB    REINFORCEMENT.  161 

the  prevention  of  diagonal  cracks  between  successive  stirrups 
although  they  may  prevent  final  failure  by  the  extension  of  a 
crack  horizontally  along  the  reinforcing  rods.  Stirrups  should 
be  looped  around  the  horizontal  rods  so  as  to  be  firmly  anchored 
at  their  lower  end  (or  upper  end  at  points  of  negative  moment), 
where  the  stress  is  a  maximum,  but  attachment  to  the  rod  is 
not  necessary,  as  the  office  of  the  stirrup  is  to  prevent  vertical, 
or  nearly  vertical,  distortion.  The  value  of  a  stirrup  unless 
anchored  or  looped  at  the  top  is  limited  by  its  strength  of 
bond,  and  as  its  length  is  not  great  this  point  may  need  con- 
sideration. In  some  tests  at  the  University  of  Wisconsin  final 
failure  has  resulted  from  slipping  of  the  stirrups.  Stirrups 
made  of  small  sections  or  bent  in  loops  are  advantageous  in  this 
respect.  Where  separate  inclined  reinforcement  is  used  there 
is  danger  of  its  slipping  along  the  horizontal  rods  if  the 
inclination  is  too  great.  'If  attached  to  the  horizontal  rods, 
however,  such  reinforcement  is  very  effective,  not  only  with 
respect  to  shear  but  also  in  increasing  the  bond  strength  of 
the  main  bars. 

Bent  rods  alone  are  apt  to  be  of  limited  value,  owing  to  the 
difficulty  of  providing  rods  close  enough  together.  Conven- 
ience of  horizontal  reinforcement  calls  for  comparatively  few 
rods  of  large  size,  which  provides  too  few  for  effective  diagonal 
reinforcement.  Where  large  rods  are  bent  up  the  length  of 
the  bent  end  should  be  made  sufficient,  by  bending  at  a  small 
angle,  to  develop  the  requisite  bond  strength.  Some  tests  of 
beams  show  failure  of  bond  in  the  case  of  short  bent  rods.  In 


t 

FIG.  51.  FIG.  52. 

the  case  of  continuous  girders  it  is  convenient  to  extend  the 
bent  rod  horizontally  at  the  top  over  the  support  to  furnish 


162 


TESTS  OF  BEAMS  AND  COLUMNS. 


[On.  IV. 


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164  TESTS    OF    BEAMS    AND    COLUMNS  [Cn.  IV 

tension  reinforcement.  A  very  satisfactory  arrangement  of 
web  reinforcement  is  a  combination  of  bent  rods  and  vertical 
stirrups,  and  especially  is  this  the  case  in  continuous-beam 
construction.  Tests  of  various  arrangements,  so  far  as  the 
authors  have  been  able  to  find,  show  the  best  results  from  this 
method  under  the  ordinary  conditions  and  proportions.  Web 
reinforcement  of  woven  wire  or  expanded  metal  should  give 
good  results. 

1 08.  Effect  of  Stirrups  on  Stress  in  the  Horizontal  rods. — 
A  careful  study  of  the    distribution    of  stress  which    exists 
after  a  beam  begins  to  rupture  on  a  diagonal  line  will  show 
the  fact  that  a  stirrup,  whether  vertical  or  inclined,  will  relieve 
the  stress  in  the  horizontal  rods  at  the  point  of  rupture.    Thus 
in  Fig.  52,  if  the  concrete  no  longer  has  tensile  strength,  the 
value  of  the  tension  T  in  the   horizontal  rods  at  the  line  of 
rupture,  if  unaided  by  the  stirrup  stress  Si  or  £2,  is  equal  to 
Vx/a,  the  same  as  its  value  was  at  section  N  before  rupture 
began.     The  moment  of  the  stress  in  the  stirrup  about  the 
point  A,  whether  the  stirrup  be  vertical  or  inclined,  serves  to 
reduce  the  value  of  T.    Without  the  stirrup  there  is  therefore 
more  danger  of  failure  of  bond  near  the  ends  of  the  beam. 

109.  Results  of  Tests. — In  Table  No.  8  are  given  in  a  classi- 
fied form  the  most  important  tests  or  rectangular  beams  which 
lend  information  on  web  stresses  and  web  reinforcement.    The 
reference  number  in  the  first  column  refers  to  the  list  of  authori- 
ties on  p.  123.    In  this  table  are  given  the  significant  facts  as 
far  as  practicable,  although  a  detailed  inspection  of  the  reports 
referred  to  is  necessary  for  a  thorough  study  of  the  tests.    The 
kind  of  failure  denoted  as  a  "shear  failure"  is  so  called  for 
convenience;   they  are  diagonal-tension  failures  brought  about 
by  large  shearing  stresses  and  hence  may  be  measured  by  the 
shearing  forces  present.     The  average  shearing  stress  on  a 
vertical  section  at  failure  is  given.    While  the  maximum  shear- 
ing stress  is  somewhat  greater  than  this  (Art.  89)  the  average 
stress  is  practically  as  good  a  standard  of  measure  and  is  much 
more  readily  calculated.    Where  the  failure  was  not  a  shear 


109.1 


WEB    REINFORCEMENT. 


165 


failure  the  figures  for  shearing  stress  are  valuable  as  indicating 
what  the  maximum  stresses  were,  although  the  beam  may 
have  withstood  still  larger  stresses  if  failure  had  not  occurred 
in  some  other  way. 

Straight  Reinforcement  Only. — The  tests  of  Professor  Talbot, 
Professor  Marburg,  and  Mr.  Harding  give  values  of  from  95 
to  123  lbs/in2  under  quite  a  variety  of  conditions.  Mr.  Carson, 
with  specially  good  concrete,  secured  values  of  about  200  in  the 
case  of  high-elastic-limit  deformed  bars  and  182  for  plain  bars, 
which,  however,  failed  in  tension.  In  the  University  of  Wiscon- 
sin tests  on  overhanging  beams,  which  represented  beams  of 
great  depth,  those  with  straight  bars  gave  a  value  of  161  lbs/in2 
and  double  reinforced  beams  values  from  155  to  194  lbs/in2, 
depending  upon  the  per  cent  of  steel  used.  Other  tests  by 
Talbot  *  in  which  straight  bars  only  were  used  gave  results 
for  average  shear  as  follows: 


Average  Shearing  Stress  (v')  at  Rupture, 

lbs/in2. 

Minimum  and 
Maximum. 

Average  of  Group. 

2 

1:2:4 

117-122 

120 

11 

1:3:5* 

71-124 

90 

4 

1:3:6 

83-92 

88 

1 

1:4:7^ 

46 

46 

4 

1:5:10 

54-64 

58 

The  amount  of  steel  varied  from  1  to  2.2%,  but  this  had 
no  apparent  effect  upon  the  strength,  all  beams  failing  in  diag- 
onal tension.  Stirrups  spaced  12  in.  apart  on  10-in.  beams 
showed  little  or  no  strengthening  effect.  Professor  Talbot 
estimates  the  maximum  shearing  strength  (which  is  about  15% 
greater  than  the  average  values  above  given)  at  J  to  J  of  the 
tensile  strength  of  the  concrete.  The  very  low  values  obtained 
in  the  case  of  the  poorer  concretes  should  be  noted. 


*  Bull.  No.  14,  Univ.  of  111.,  1907. 


166  TESTS    OF    BEAMS  AND   COLUMNS.  [Cn.  IV. 

As  stated  in  Art.  105  the  amount  of  horizontal  steel  has  a 
direct  bearing  on  shear  failures  for  the  reason  that  large  areas 
of  steel  with  low  unit  stresses  permit  less  extension  of  the  con- 
crete than  small  areas  with  high  working  stresses.  This  effect 
is  shown  in  a  marked  manner  in  a  series  of  tests  made  at  the 
University  of  Wisconsin  on  small  mortar  beams  of  1  •  3  mixture. 
The  beams  were  3"X4J/r  in  cross-section  and  4  ft.  span  length. 
Loads  were  applied  at  two  points  a  varying  distance  apart. 
Only  straight  reinforcement  was  used,  amounting  to  1.41%. 
The  tensile  strength  of  the  material  was  high,  being  490  lbs/in2. 
The  results  were  as  follows : 

Distance  Apart  Average  Shearing 
of  Loads.  Stress. 

Inches.  Lbs/in2. 

Centre  Load.  177 

12  200 

24  220 

32  316 

36  512 

40  850 

44  1035 

The  increase  in  strength  as  the  loads  approached  the  supports 
must  be  due  largely  to  the  decrease  in  moment  stress  and 
consequent  distortion,  which  is  essentially  what  occurs  when 
large  areas  of  steel  and  low  working  stresses  are  used. 

Beams  with  Web  Reinforcement.— MX.  Harcling's  tests  included 
only  bent  rods,  and  with  these  very  considerably  higher  ulti- 
mate values  were  obtained  than  for  straight  rods,  averaging  for 
the  three  groups  190  lbs/in2.  Plain  bars,  bent,  gave  tension 
failures,  these  bars  being  of  lower  elastic  limit  than  the  deformed 
bars.  These  results  are  therefore  of  negative  value.  In  some 
of  Mr.  Harcling's  tests  the  inclined  bars  pulled  out,  the  bent 
ends  being  relatively  short,  as  indicated  in  the  sketches.  An 
inspection  of  the  deflection  curves  of  these  beams  will  show 
that  those  in  which  the  rods  were  not  bent  were  the  stiffer 
beams,  owing  to  the  greater  average  amount  of  steel  carrying 


§  no.]  TESTS    ON    T-BEAMS.  167 

the  bending  moment.  Mr.  Carson's  results  average  227 
lbs/in2  for  curved  bars  and  from  220  to  338  lbs/in2  for 
straight  bars  with  stirrups,  the  strength  increasing  with  in- 
creasing per  cent  of  metal.  The  stirrups  were  1"  X  \"  straps 
spaced  about  7  in.  apart.  Tests  by  Talbot  in  which  curved 
and  inclined  rods  were  used,  but  in  which  no  rods  continued 
straight  for  the  entire  length  of  the  beam,  showed  results 
very  little  better  than  for  straight  rods.* 

A  reference  to  Table  No.  13  will  show  the  effect  of  stirrups 
on  the  ultimate  strength  and  method  of  failure  of  beams  rein- 
forced for  compression.  In  the  tests  of  Mr.  Withey  the  bent 
rods  alone  gave  258  lbs/in2  and  stirrups  alone  average  240, 
while  the  combination  gave  334,  with  a  tension  failure,  show- 
ing still  greater  web  stresses  possible.  Expanded  metal,  as 
used,  proved  too  weak,  as  it  pulled  apart  at  a  shearing  value 
of  240  lbs/in2.  T-beam  tests  described  in  Art.  110  indicate 
that  a  value  of  350  lbs/in2  may  readily  be  reached  with  stiri 
and  bent  rods  even  with  a  relatively  poor  concrete, 

The  importance  of  tensile  strength  in  the  concrete  should 
be  noted  in  this  connection,  as  the  diagonal  tension  or  shear 
failure  is  the  one  to  be  most  feared  and  therefore  most  care- 
fully guarded  against. 

no.  Tests  on  T-Beams. — The  reinforcing  of  T-beams  re- 
quires special  care  in  providing  against  shearing  stresses.  Where 
a  floor  slab  forms  the  upper  part  of  a  beam  there  will  usually 
be  ample  strength  in  compression  for  any  depth  likely  to  be 
selected.  The  design  of  the  stem  of  the  T,  or  the  beam  below 
the  slab,  is  therefore  largely  a  question  of  providing  sufficient 
concrete  and  reinforcement  to  take  care  of  the  shearing  stresses. 
In  this  case,  therefore,  it  is  important  to  provide  a  strong  web 
for  shearing  stresses,  as  the  strength  in  this  respect  will  com- 
monly determine  its  size.  In  Tables  Nos.  9  to  HA  are  given 
the  most  important  tests  on  T-beams  known  to  the  authors. 
The  percentage  of  steel  is  calculated  with  reference  to  a  rectan- 


*  Bull.  No.  14,  Univ.  of  111.,  1907. 


168 


TESTS    OF    BEAMS    AND    COLUMNS. 


[Cn.  IV, 


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§  110,]  TESTS    ON    T-BEAMS.  169 

gular  beam  having  a  cross-section  equal  to  the  circumscribing 
rectangle. 

The  yield  point  of  the  plain  steel  in  the  tests  of  Table 
No.  9  was  about  37,000  lbs/in2,  and  its  ultimate  strength 
51,000  lbs/in2.  A  load  of  about  19,000  Ibs.  would  stress  the 
steel  in  the  beams  having  .84%  reinforcement  to  the  yield 
point.  This  limit  is  exceeded  only  in  the  last  three  of  the  list. 
In  these  beams  inclined  stirrups  were  used,  placed  in  a  notch 
in  the  bar,  in  all  other  series  the  stirrups  were  placed  vertically. 

Reviewing  these  experiments  we  note,  first,  the  results  with 
straight  bars  and  no  stirrups.  The  beams  having  the  .8-in. 
rods  and  the  Thacher  bars  developed  a  value  of  119  lbs/in2 
average  shearing  stress,  while  the  .4-in.  rods  developed  198, 
the  difference  being  due  doubtless  to  the  difference  in  bond 
strength,  although  the  previous  experiments  cited  would  indi- 
cate that  not  much  greater  value  than  the  latter  figure  could 
be  expected  from  straight  bars  only. 

Noting  the  next  five  beams,  all  have  straight  rods  and 
vertical  stirrups,  No.  4  having  stirrups  spaced  8"  apart,  while 
the  others  have  a  spacing  of  4"  or  less  near  the  support.  For 
the  former  a  value  of  205  lbs/in2  was  reached,  while  the  three 
others  averaged  341,  all  being  nearly  the  same  despite  the 
variety  of  bars  used.  No.  9  had  bent  bars  and  no  stirrups, 
giving  a  strength  of  252,  while  No.  10  had  bent  rods  and  stirrups 
rather  widely  spaced,  developing  334.  Nos.  11-14  had  inclined 
stirrups  attached  to  the  bars  and  all  but  the  first  gave  high 
values  of  over  450  for  the  shear. 

In  these  tests  it  should  be  noted  that  a  load  of  16,000  Ibs. 
would  develop  in  the  rods  a  theoretical  stress  of  (8000  x20)/4.2  = 
38,000  lbs/in2.  For  the  .8-inch  rods  this  would  require  an 
average  bond  strength  of  about  38,000/(2Xf  X?rX25i)  = 
320  lbs/in2,  about  all  that  could  be  expected.  The  .4-inch 
rods  would  be  stressed  one-half  as  much  in  bond.  The  spacing 
of  stirrups  in  No.  10  was  too  great  to  be  entirely  efficient.  The 
inclined  attached  stirrups  gave  the  best  results  in  these  tests, 
but  whether  similar  results  would  be  obtained  where  strength 


170 


TESTS    OP    BEAMS    AND   COLUMNS. 


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TESTS    ON    T-BEAMS 


171 


TABLE  No.  11. 
T-BEAM  TESTS  AT  THE  UNIVERSITY  OF  ILLINOIS  * 

Concrete,  1:2:4;  age  about  60  days;  comp.  strength  of  cubes=1820 
lbs/in2. 

Steel:  yield  point  of  plain  round  =  38300  lbs/in2;  of  Johnson  bars= 
53800  lbs/in2. 

Size  of  beams:  thickness  of  flange  =3|  in.;  thickness  of  web  =  8  in.; 
depth  to  center  of  steel=10  in.;  total  length=ll  ft.;  span  length=10  ft.; 
width  of  flange  varied. 

Stirrups:  made  of  J-in.  Johnson  bars;  five  stirrups  at  each  end  spaced 
6  in.  apart. 

Loads  applied  at  third  points.     All  failures  were  steel  tension  failures. 


Number  and  Size 
of  Rods. 

Average 

Num- 
ber. 

Width  of 
Flange. 
Inches. 

Percent- 
ape 
Reinforce- 
ment. 

Total 
Breaking 
Load. 
Pounds. 

Shearing 
Stress  on 
Section 
8"X10" 

Stress  in 
Steel. 
lbs/in2. 

Ibs/in2. 

1 

] 

1.05 

3^ 

"  Johnson 

46700 

293 

64300 

4 

16 

1.10 

41 

"  Plain  round 

32410 

203 

41500 

7 

j 

1.10 

4i 

//       «         <  i 

30100 

188 

38100 

3 

1 

0.93 

4j 

"  Johnson 

55700 

347 

57500 

6 

24 

0.92 

/5 

1(2 

"  Plain  round 
bars  bent  up) 

39300 

246 

40700 

8 

0.92 

hi 

1(2 

"  Plain  round 
bars  bent  up) 

40100 

250 

41200 

2 

1.05 

61 

-"  Johnson 

80500 

503 

55700 

5 

>•      32 

1.05 

f«i 

1(2 

"  Johnson 
bars  bent  up) 

83300 

521 

57400 

9 

0.97 

[7! 

1(3 

"  Plain  round 
bars  bent  up) 

50900 

318 

37600 

of  bond  was  not  in  question  cannot  be  stated.  In  case  of  weak 
bond  an  attached  inclined  stirrup  virtually  adds  much  to  the 
bond  strength  of  the  bar. 

In  Table  No.  10  are  given  further  results  of  tests.  In  the 
first  four  tests  the  supports  "were  placed  too  near  the  ends  of 
the  beam  (4  inches)  with  the  result  that  after  the  initial  crack- 
ing the  bars  soon  pulled  out.  After  reducing  the  span  length 
to  5  feet  no  further  trouble  in  this  respect  was  experienced. 
The  results  correspond  closely  with  those  given  in  the  other 
tables. 


*  Bulletin  No.  12,  Eng.  Exp.  Station,  Univ.  of  111  ,  1907. 


172  TESTS   OF   BEAMS  AND   COLUMNS.  [Cn.  IV. 

Table  No.  11  contains  results  of  recent  tests  by  Professor 
Talbot.  The  maximum  values  of  shearing  stress  are  unusually 
high  and  indicate  very  effective  web  reinforcement.  As  no 
shear  failures  occurred  the  possible  limit  of  strength  of  web 
was  not  determined.  The  excess  of  stress  in  the  steel  as  com- 
pared to  the  yield  points  should  be  noted,  due  in  large  measure 
no  doubt  to  the  excessive  compressive  strength  of  the  flange 
and  the  thorough  web  reinforcement. 

Table  No.  HA  contains  results  of  additional  tests  on  T- 
beams  made  at  the  University  of  Wisconsin.  In  these  tests 
the  yield  point  of  the  corrugated  bars  was  about  48,000  lbs/in2 
and  that  of  the  f-in.  rods  was  41,000  lbs/in2.  These  limits 
correspond  closely  to  the  stresses  in  the  steel  at  failure,  except- 
ing in  the  case  of  beams  GI  and  G2  which  failed  by  shear.  The 
table  contains  results  of  value  with  respect  to  shearing  stresses 
and  the  use  of  stirrups  and  bent  rods  for  shear  reinforcement. 
In  the  progress  of  the  tests  the  occurrence  of  the  first  diagonal 
crack  was  carefully  noted  and  the  maximum  shearing  stress 
at  this  load  is  calculated  and  given  in  the  table.  It  will  be 
noted  that  there  is  a  fairly  close  agreement  between  this  value 
and  the  tensile  strength  of  the  concrete  as  given  in  the  next 
column.  The  average  value  for  the  maximum  shearing  stress 
is  179  lbs/in2  whereas  the  average  tensile  strength  is  187  lbs/in2. 
This  would  indicate  that  in  spite  of  stirrups  the  concrete  is 
likely  to  open  up  at  a  diagonal  tensile  stress  about  equal  to  its 
usual  tensile  strength.  The  table  also  gives  the  shearing 
stress  at  ultimate  load,  both  the  average  and  maximum  stress 
(at  neutral  axis  and  below)  being  given.  The  web  reinforce- 
ment was  effective  in  preventing  shear  failures,  excepting  in 
the  case  of  beams  GI  and  G%  where  no  bent  rods  were  used, 
in  all  other  cases  the  web  reinforcement  was  ample  and  hence 
no  conclusions  can  be  drawn  as  to  relative  value  of  the  different 
kinds  of  reinforcement.  The  wire  mesh  gave  good  reinforce- 
ment and  was  convenient  to  use.  In  beams  GI  and  G2  failure 
by  diagonal  tension  occurred,  the  stirrups  breaking  at  the 
maximum  load.  These  tests  indicate  that  by  the  use  of  bent 


§  no.] 


TESTS   OF   T-BEAMS. 


173 


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174  TESTS    OF    BEAMS    AND    COLUMNS.  [Cn.  IV. 

rods  and  stirrups  an  average  shearing  strength  of  400  lbs/in2 
can  readily  be  developed  with  good  concrete. 

in.  Conclusions  as  to  Shearing  Strength. — From  the  avaiU 
able  data  it  would  appear  that  with  ordinary  concrete  and  no 
web  reinforcement  the  ultimate  average  shearing  strength  is 
about  100  lbs/in2  and  that  this  strength  can  readily  be  in- 
creased by  the  use  of  web  reinforcement  to  300  to  400  lbs/in2. 
The  latter  figure  may,  from  our  present  knowledge,  be  taken  as 
about  the  maximum  value  with  ordinary,  closely  spaced  web 
reinforcement.  It  appears  also  that  the  shearing  strength  of  a 
T-beam  is  about  the  same  as  that  of  a  rectangular  beam  of  the 
same  depth  and  a  width  equal  to  the  width  of  the  stem  of  the 
T.  It  is  to  be  understood  that  the  shearing  stress  is  here  used 
merely  as  a  convenient  measure  of  the  diagonal  tensile  stress, 
which  is  really  the  stress  involved.  This  being  the  case  it  would 
be  incorrect  to  take  any  account  of  the  shearing  strength  of 
the  steel  in  designing  the  reinforcement,  as  is  sometimes  done. 

112.  Beams  Reinforced  for  Compression.  —  Generally 
speaking,  it  is  more  economical  to  carry  compressive  stresses 
by  concrete  than  by  steel,  but  limitations  as  to  size  sometimes 
makes  it  desirable  to  strengthen  the  compressive  side  of  a  beam. 
In  cases,  also,  where  both  positive  and  negative  moments  exist 
in  the  same  beam,  either  as  alternating  stresses  or  simultane- 
ously at  different  points,  steel  reinforcement  will  be  used  on 
both  sides  and  its  value  on  the  compressive  side  needs  to  be 
known.  Obviously,  steel  reinforcement  on  the  compression 
side  will  have  little  effect  in  beams  that  would  otherwise  fail 
in  tension  or  shear,  although  there  would  be  some  gain  owing 
to  increased  distance  between  centers  of  tensile  and  compres- 
sive forces.  The  effectiveness  of  steel  in  compression  has  some- 
times been  questioned,  but  the  results  of  tests  on  beams  and 
columns  indicate  that,  in  ordinary  ratios  at  least,  the  steel 
does  its  share  of  work. 

Table  No.  12  gives  results  of  tests  on  double  reinforced 
beams  made  at  the  University  of  Wisconsin.  No  stirrups 
were  used  in  the  first  eight  beams  and,  as  a  consequence,  all 


J  112.] 


BEAMS    REINFORCED    FOR   COMPRESSION. 


175 


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176  TESTS    OF    BEAMS    AND    COLUMNS.  [Cn.  IV. 

these  beams,  except  the  first  two,  failed  by  diagonal  tension. 
The  last  four  beams  were  supplied  with  stirrups  looped  around 
the  lower  bars  and  consisting  of  J-in.  round  steel,  ten  stirrups 
being  used  at  each  end  between  the  load  and  support.  In  all 
beams  two  of  the  bottom  rods  were  bent  up  at  about  2  feet 
from  the  end  and  one  bar  at  4  feet  from  the  end.  Fig.  52a 
shows  arrangement  of  reinforcement  and  location  of  cracks  in 
beam  W\.  They  are  numbered  in  the  order  of  occurrence, 
crack  "F"  being  the  point  of  final  failure,  the  steel  passing  its 
yield  point. 

The  neutral  axis  was  found  by  the  use  of  extensometers, 
after  which  the  stresses  in  steel  and  concrete  at  "load  con- 
sidered" were  found,  assuming  the  compression  in  the  con- 
Beam  w, 


"/••  -    ^ 


A  n 

FIG.  52a. 

crete  to  follow  the  parabolic  law.  The  tensile  stresses  in  the 
steel  as  calculated  by  the  two  methods  agree  very  closely. 
The  compression  in  the  concrete  is  determined  by  subtracting 
from  the  total  compression  the  compressive  stress  in  the  steel. 
The  crushing  strength  of  corresponding  compression  speci- 
mens is  also  given  in  column  8,  and  in  the  following  column 
the  ratio  of  the  calculated  stress  in  the  beam  to  the  crushing 
strength.  Except  in  the  case  of  the  first  two  beams  the  ulti- 
mate compressive  strength  of  the  concrete  was  not  reached, 
although  in  the  last  four  beams  crushing  took  place  very  soon 
after  the  maximum  load  was  reached.  In  the  next  column 
are  given  the  ratios  of  calculated  stresses  in  compression  steel 
and  in  concrete.  These  are  in  fair  agreement  with  the  value 
of  n  for  concrete  at  rupture.  The  concrete  in  beams  W  hap- 
pened to  be  especially  good  and  in  beams  X  it  was  poor.  The 
stirrups  used  in  the  last  four  beams  proved  effective  in  prevent- 


112.1 


BEAMS   REINFORCED   FOR    COMPRESSION. 


177 


ing  shear  failures.  The  results  of  these  tests  indicate  that  the 
steel  is  taking  its  share  of  stress  and  that  the  compression  side 
of  the  beam  is  strengthened  in  accordance  with  the  usual 
theory.  Obviously,  in  order  to  secure  full  benefit  of  the  steel 
up  to  rupture,  a  fairly  high  elastic  limit  material  should  be  used. 
The  tests  show,  further,  that  the  compressive  strength  of  con- 
crete in  a  beam  is  fully  equal  to  and  probably  somewhat  greater 
than  the  strength  as  determined  from  the  usual  compression 
test. 

Fig.  53  gives  a  typical  set  of  curves  for  double  reinforced 
beams.  Beams  T2  and  TF2  had  equal  amounts  of  compressive 
reinforcement,  but  beam  TF2  was  provided  with  stirrups  while 
T2  was  not.  These  conditions  resulted  in  a  shear  failure  in 
the  former  case  and  a  tension  failure  in  the  latter.  Com- 
paring with  those  shown  in  Art.  100,  it  will  be  seen  that  these 
beams  are  much  stiffer  and  apparently  more  perfectly  elastic, 
as  would  be  expected  from  the  nature  of  the  reinforcement. 

TABLE  No.  13. 
TESTS  OF  BEAMS  REINFORCED  FOR  COMPRESSION. 

BOSTON  TRANSIT  COMMISSION.* 

Beams  and  material  as  described  in  Table  No.  8.  All  beams  reinforced 
with  \  '  corrugated  bars,  with  same  number  top  aid  bottom.  Stirrups 
I"  X  J".  spaced  about  7".  Centre  loads. 


Total  Rein- 

Num- 
ber. 

forcement. 

Use  of 
Stir- 
rups. 

Load  at 
First 
Sign  of 
Failure. 
Pounds 

Ulti- 
mate 
Load. 
Pounds. 

M 
bd2 

Average 
Shearing 
Stress, 
t1'. 
Lbg/in2. 

Kind  of 
Failure. 

Num- 
ber of 

Per- 

Bars. 

centage 

72 

4 

vr_ 

9920 

10980 

513 

126 

Tension 

78 

4 

1/?o 

INO 

11424 

14148 

660 

162 

'  < 

71 

4 

.62 

Voo 

11000 

16506 

766 

188 

Shear  &  tens. 

77 

4 

x  es 

11224 

15072 

701 

172 

«        ><     « 

70 

6 

•VT 

14992 

16096 

740 

182 

Shear 

76 

6 

9   dd  i 

INO 

16716 

17300 

796 

195 

«  e 

69 

6 

Z  .  ft  * 

"\T 

17724 

23972 

1106 

272 

Tension 

75 

6 

Yes 

14476 

21284 

990 

244 

t  ( 

68 

8 

XT 

19044 

19044 

880 

215 

Shear 

74 

8 

3OC 

No 

17200 

18584 

854 

210 

t  ( 

67 

8 

.25 

Voo 

21200 

30168 

1400 

344 

Tension 

73 

8 

! 

les 

22132 

29178 

1347 

332 

«  « 

*  Tenth  Annual  Report,  1904. 


178 


TESTS    OF   BEAMS   AND    COLUMNS. 


CH.  IV. 


ow 
500 
400 
300 
200 
100 

0 
600 

500 
|  400 
£  300 

09 

I  200 

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5,800 
jj  700 
600 
500 
400 
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Top  Reinforcement  :        none 

Y 

2 

/ 

/ 

// 

^ 

7 

/£j& 

/ 

^ 

b« 

A 

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Top  Reinforcement        =.  1.0  # 

| 

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f 

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Top  Reinforcement 

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2      0 

Defl 
3     0 

ection 
4      0 

sin  Ii 
5      0 

iches 
6      0 

7      0 

8     0 

9      1 

0 

.0005             .0001            .0015            .0080             .0025             .0030 

Deformation  per  Unit  Length 

FIG.  53. 


§112,  TESTS  OF  COLUMNS.  179 

Table  No.  13  gives  results  of  tests  on  double-reinforced  beams 
by  the  Boston  Transit  Commission.  The  table  is  of  value 
mainly  in  showing  the  benefit  of  stirrups.  Crushing  failures 
were  obtained  in  but  few  cases  in  this  series  of  tests,  even  where 
no  compressive  reinforcement  was  used,  so  that  little  advantage 
could  be  expected.  It  should  be  noted  that  where  stirrups  are 
not  used  the  results  shown  in  this  table  are  very  nearly  the 
same  as  those  of  Table  No.  12,  although  the  quality  of  the 
concrete  in  the  latter  case  was  much  inferior.  Conditions  were 
such  that  the  full  strength  of  the  concrete  was  not  developed  in 
the  tests  of  Table  No.  13. 

H2a.  Experiments  on  Deflections  of  Beams. — Probably  the 
most  complete  and  accurate  deflection  measurements  ever  made 
are  those  by  Bach.  About  50  rectangular  and  20  T-beams 
are  reported  on  by  him.*  The  rectangular  beams  were  2  m. 
long,  30  cm.  deep  and  15,  20,  or  30  cm.  wide.  They  were 
reinforced  with  a  single  straight  rod,  several  straight  rods, 
straight  rods  with  stirrups,  or  several  rods,  some  bent  up; 
the  percentages  varied  from  about  .4  to  1.35.  The  T-beams 
were  3  m.  long,  45  cm.  wide,  48  cm.  deep,  flange  10  and  web 
20  cm.  thick.  They  were  reinforced  with  straight  rods,  with 
or  without  stirrups,  or  rods,  some  bent  up,  with  and  without 
stirrups;  the  percentage  of  steel  was  about  .8  in  all  of  them. 
The  beams  rested  on  end  supports,  and  were  loaded  at  the 
quarter  or  third  points.  They  were  made  in  sets  of  three  (as 
nearly  alike  as  possible)  and  the  load-deflection  curves  for  any 
set  are  in  remarkably  good  agreement.  Deflections  were 
measured  at  5  or  7  different  points  along  the  beam  and  to 
the  nearest  .005  mm. 

Fig.  53a  shows  the  load-deflection  curves  for  4  sets  of  beams. 
Groups  1  and  2  relate  to  rectangular  beams;  in  (1)  the  beams 
were  15  cm.  wide,  reinforced  with  .5%  of  steel  in  3  rods,  2 
bent  up  at  each  end;  in  (2)  the  beams  were  20  cm.  wide  and 


*  Mitteilungen  iiber  Forschungsarbeiten  auf  dem  Gebiete  des  Ingenieur- 
wesens,  Hefte  39,  45,  46,  47  (1907). 


180 


TESTS    OF   BEAMS   AND   COLUMNS.  [  CH.  IV. 


I 


w 


7 


z 


I, 


Deflections.-  Scale:  1  in.=0.2  mm, 
FIG.  53a 


§  1126.]     EXPERIMENTS  ON    DEFLECTIONS    OF  BEAMS.  181 

reinforced  with  1.35%  of  steel  in  3  rods,  2  bent  up  at  each 
end.  Groups  3  and  4  relate  to  T-beams;  in  the  beams  of  (3) 
there  were  3  straight  rods  (.8%)  and  24  stirrups,  and  in  those 
of  (4)  there  were  5  rods,  4  bent  up  at  each  end,  (.87%)  and 
24  stirrups.  Only  a  part  of  each  curve  is  given.  The  dot  on 
each  corresponds  to  one-quarter  ul  timateload;  dots  on  exten- 
sions of  group  4  would  be  a  trifle  higher  than  in  group  3. 

The  dashed  lines  are  graphs  of  the  deflection  formulas  (see 
Arts.  92d  and  92e)  corresponding  to  the  various  beams,  n 
having  been  taken  as  equal  to  8  for  reasons  given  in  the  next 
article.  The  deflection  formula  agrees  as  well  with  other  sets 
in  Bach's  tests  except  in  a  few  cases  in  which  the  reinforce*, 
ment  consisted  of  a  single  straight  rod  and  stirrups. 

Deflection  measurements  on  beams  tested  in  America  seem 
not  to  have  been  made  with  special  care,  as  there  is  consider- 
able discordance  in  the  published  results.  Among  the  best 
are  some  reported  by  Talbot,  a  few  *  of  which  are  represented 
in  Fig.  536.  Groups  1  and  2  relate  to  two  sets  of  T-beams, 
12  in.  deep  (over  all),  flange  3}  and  web  8  in.  thick;  the  span 
was  10  ft.,  and  loads  at  third  points.  The  three  in  group  1 
were  16  in.  wide,  reinforced  with  straight  bars  (about  1%) 
and  stirrups,  the  three  in  group  2  were  24  in.  wide,  reinforced 
as  others  except  some  rods  were  bent  up.  Curve  3  is  for  a 
very  large  rectangular  beam;  its  breadth  was  25  in.,  depth  to 
steel  30.5  in.,  span  23.5  ft.,  and  percentage  of  steel  1.25.  Only 
a  portion  of  each  curve  is  shown;  the  dot  on  each  corresponds 
to  one-fourth  the  ultimate  applied  load.  The  dashed  lines  are 
the  graphs  of  the  deflection  formulas  (Arts.  92d  and  92e)  for 
the  corresponding  beams;  8  is  the  value  of  n  used  in  groups  1 
and  2. 

1 1 25.  Stirrups  and  Bent-up  Rods  do  not  affect  the  stiffness 
of  the  beam  materially  for  working  loads ;  but  they  do  increase 
the  ultimate  deflection  as  well  as  strength.  Bach's  tests 
clearly  show  this  to  be  true,  for  example: 

*  Bulletin  Univ.  of  111.  Eng.  Exp.  Station,  No.  12  (1907);   Eng.  News, 
Vol.  LX,  p  145  (1908). 


182 


TESTS    OF    BEAMS    AND  COLUMNS.  [Cn.  IV. 


I 


zz 


/ 


Deflections,  for  1  and  8,1  in.-0.05  in;  for  3, 1  in=0.2  in. 
FIG.  536. 


§  1126.]       EXPERIMENTS  ON   DEFLECTIONS  OF  BEAMS. 


183 


(1)  Column  a  of  the  adjoining  table  (No.  13A)  gives  the 
average  deflections  for  three  beams  (numbers  7,  13,  and  14) 
corresponding  to  the  loads  tabulated;  the  beams  were  rein- 
forced with  a  single  straight  rod  (p  about  .9%).  Column  b 
gives  the  average  deflections  for  another  set  of  three  (29,  32, 
and  37) ;  these  were  reinforced  like  the  first  set  but  with  sixteen 
stirrups  added.  The  fourth  column  gives  the  percentage 
differences  between  the  deflections  of  the  two  sets  of  beams 
up  to  4000  kg.  The  average  ultimate  deflections  of  the  two 
sets  were  1.78  and  2.3  mm.,  and  the  ultimate  loads  18,900 
and  23,250  kg.  respectively. 

TABLE  No.  13A. 
DEFLECTIONS  OF  RECTANGULAR  BEAMS. 


Deflection  (millimeters). 

Load, 

(Kilos). 

a. 

b. 

Diff. 

A. 

B. 

Diff. 

500 

.052 

.052 

0% 

.048 

.050 

+  4.0% 

1000 

.110 

.107 

-2.7 

.107 

.110 

+  2.8 

1500 

.175 

.165 

-5.7 

.167 

.173 

+  3.6 

2000 

.245 

.232 

-4.9 

.232 

.248 

+  6.8 

2500 

.322 

.307 

-4.6 

.308 

.330 

+  6.9 

3000 

.428 

.417 

-2.6 

.403 

.403 

+  6.7 

3500 

.608 

.580 

-4.6 

.538 

.585 

+  8.7 

4000 

.793 

.767 

-3.3 

.775 

.902 

+  14.0 

(2)  Column  A  of  the  same  table  gives  the  average  deflections 
for  a  set  of  beams  (40,  43,  and  45)  which  were  reinforced  with 
three  straight  rods  (p  =  .55%);   and  B  the  average  deflections 
for  a  set  (49,  51,  and  53)  reinforced  like  the  first,  but  two  of  the 
rods  were  bent  up  at    each  end.     The  last  column  gives  the 
percentage  differences  between  the  average  deflections  of  the 
two  sets  of  beams.     The  average  ultimate  deflection  of  sets  A 
and  B  were  3.38  and  3.45  mm.  and  their  average  ultimate 
loads  8250  and  8600  kg.  respectively. 

(3)  The  numbered  columns  in  the  adjoining  table  (No.  13B) 
give  the  average  deflections  of  six  sets  of  T-beams,  three  in 


184 


TESTS    OF    BEAMS    AND    COLUMNS. 


[Cn.  IV. 


TABLE  No.  13B. 
DEFLECTIONS  OF  T-BEAMS. 


Loads 
(kilos). 

Deflections  (millimeters). 

1 

2 

3 

4 

5 

6 

Diff. 

2000 
4000 
6000 
8000 
10000 

.090 
.193 
.307 

.087 
.183 
.290 

.088 
.180 
.287 
.398 

.093 
.190 
.303 
.422 

.092 
.190 
.293 
.407 
.553 

.095 
.192 
.303 
.420 
.563 

?:f* 

10.5 
6.0 
1.8 

.435 
.577 

.400 
.535 

.568 

.553 

each  set,  for  the  loads  tabulated.  The  beams  were  alike 
except  as  to  reinforcement.  Beams  of  set  1  were  reinforced 
with  three  straight  rods  (p  =  .8%);  set  2  like  1  and  24  stirrups; 
set  3  like  1  and  48  stirrups;  set  4  with  five  rods  (p=.87%), 


4,600,000 


100 


200  300  400  500 

Values  of  unit  stress  fc, 

FIG.  53c. 


600 


800 


four  bent  up  at  each  end*  set  5  like  4  and  24  stirrups;  and 
set  6  like  5  except  that  a  hook  was  formed  at  each  end  of  the 
fifth  rod.  The  horizontal  lines  in  the  table  are  drawn  to 
correspond  to  one-quarter  ultimate  loads.  The  last  column 


§  113.]  TESTS    OF    COLUMNS.  185 

of  the  table  gives  the  greatest  percentage  difference  for  the 
various  working  loads.  The  average  ultimate  deflections  were 
2.4,  3.2,  3.8,  6.0,  5.8,  and  9.4  mm. ;  the  average  ultimate  loads 
23,000,  30,500,  37,800,  33,300,  41,000,  46,000  kg.  respectively. 
All  average  ultimate  deflections  are  not  reliable. 

H2C.  On  the  Value  of  n  for  Deflection  Formulas.— As 
explained  in  Arts.  92c  and  b,  the  value  of  the  modulus  of 
elasticity  to  be  used  in  deflection  formulas  should  correspond 
not  to  the  greatest  unit  stress  in  the  concrete  but  to  a  fair 
average  of  the  unit  stresses  at  all  points  in  the  beam.  Fig. 
53c  shows  how  the  secant  modulus  varied  in  four  compression 
specimens  representing  the  concrete  of  the  Bach  beams  referred 
to  in  Art.  112a.  It  was  a  1 :4  gravel  concrete  and  the  specimens 
were  about  eight  months  old  when  tested.  The  curves  show 
that  .for  unit  stresses  as  high  as  600  lbs/in2  the  moduli  averaged 
over  3J  million  (n  =  9),  and  for  the  fair  average  unit-stress  in 
Bach's  beams  under  working  loads  n  would  be  about  8. 


COLUMNS. 

113.  Tests  of  Plain  Concrete  Columns. — An  important 
series  of  tests  on  columns  are  those  made  at  the  Watertown 
Arsenal,  and  reported  in  Tests  of  Metals,  1904,  and  subsequent 
volumes.  The  principal  results  on  plain  concrete  are  given 
in  Table  No.  14. 

These  tests  indicate  an  average  strength  for  1:2:4  concrete 
of  1600  to  1700  lbs/in2,  with  no  excessive  variation  in  indi- 
vidual tests.  For  the  weaker  mixture,  1:3:6,  the  individual 
tests  are  much  more  at  variance,  indicating  greater  unrelia- 
bility. The  great  strength  of  very  rich  mortars  is  noteworthy, 
and  this  fact  is  borne  out  by  experiments  on  columns  slightly 
reinforced.  Considering  relative  cost,  a  rich  mortar  may  often 
be  the  more  advtantageous. 

Table  No.  14A  gives  results  of  tests  by  Professor  Talbot 
on  plain  concrete. 


186 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV 


TABLE  No.  14. 

TESTS  OF  PLAIN  CONCRETE  COLUMNS. 
WATERTOWN  ARSENAL,  1903-1905. 

All  columns  were  8  ft.  high  and  ranged  from  10  in.  in  diameter  to  12  in. 
square.     The  age  of  the  concrete  ranged  from  5  to  8  months. 


Crushing  Strengtl 

i,  Lbs/in2. 

Kind  of  Concrete. 

Results  of  Indi- 
vidual Tests.* 

Average 
Crushing 
Strength. 

1  :  1  mortar  

/5011+             \ 

4665 

1-2       " 

\  4320              J 
3652     2488 

3070 

1-3       " 

2062     2692 

2377 

1-4       " 

f  1564     1471  \ 

1362 

1-5       '<              

(  1050             J 
1038     1082 

1060 

1-1-2  (pebbles)  

1525     1720 

1622 

1  :  1  •  2  (trap  rock)  

3900 

3900 

1*2'  4  (pebbles) 

1506     1710 

1608 

l'2-4  (trap-rock) 

1750     19901 

1718 

1-3-6  (pebbles) 

1413              f 
462       700  \ 

807 

1  *  3  •  6  (trap-rock)   . 

1260              J 
/  1350              1 

1182 

1*2'  4  (cinders)  

\    750     1446  / 

871 

871 

1:3:6  (cinders)  

(  1060              1 

879 

\    698              J 

*  Where  two  lines  of  values  are  given,  those  in  the  first  line  are  results  obtained  in 
the  1904  series,  those  in  the  second  line  are  from  the  1905  series. 

In  general,  it  was  found  that  the  richer  mixtures  tended 
to  fail  by  true  shear  failures,  while  the  poorer  mixtures  gener- 
ally failed  by  gradual  crushing.  The  very  superior  results 
obtained  on  the  1:1J:3  mixture,  as  compared  to  the  1:2:4 
mixture,  or  poorer,  should  be  no'ted.  It  shows  the  value  of 
the  use  of  rich  mixtures  for  columns,  the  increase  in  strength 
over  the  1:2:4  concrete  being  about  32%  while  the  increase 
in  cost  would  not  be  over  10  or  15%.  Compared  to  the  results 
of  Table  No.  14  these  results  agree  as  closely  as  could  be 
expected.  The  great  variation  in  individual  tests  in  Table 
No.  14A  should  be  noted,  the  results  for  group  2  varying  from 
33%  below  to  27%  above  the  average.  Results  of  comparative 


5  113.] 


TESTS    OF    COLUMNS. 


187 


TABLE  No.  HA. 
TESTS  OF  PLAIN  CONCRETE  COLUMNS. 

UNIVERSITY  OF  ILLINOIS,  1907  * 
All  columns  were  12  in.  in  diameter  by  10  ft.  long. 


Crushing  Strength,  Lbs/in2. 

Group. 

Col.  No. 

Kind  of 
Concrete. 

Age  of 
Specimen, 
Days. 

Individual 

Average  for 

Tests. 

Group. 

1 

r  in 

{   112 

1:1£:3 

2120 
2480 

2300 

1    66 
/    62 

101 

1165 

58 

102 

2000 

69 

103 

2210 

65 

2 

104 

1:2:4 

1590 

1740 

64 

105 

1945 

62 

108 

1460 

72 

109 

1810  ' 

64 

3 

f   116 
1   117 

1:3:6 

955 
1110 

1030 

I   61 
/   62 

4 

/   121 
\   122 

1:4:8 

575 
575 

575 

\   63 
{   63 

110 

1925 

1   203 

128 

1845 

194 

5 

129 
163 

1:2:4 

1770 
2680 

2025 

181 

187 

164 

2160 

187 

168 

1770 

201 

6 

f   21 
{    22 

1:2:3! 

2650 
2770 

2710 

\  12  mo. 
/  16  mo. 

tests  on  short  cylinders  of  1:2:4  concrete,  stored  in  damp 
sand  for  9  to  11  months,  gave  an  average  crushing  strength  of 
2650  lbs/in2;  tests  on  12-inch  cubes  stored  in  air  at  age  of 
60  days  gave  an  average  value  of  about  1950  lbs/in2,  and  at 
age  of  about  200  days,  of  2350  lbs/in2. 

Other  tests  on  plain  concrete  are  given  in  Tables  Nos. 
17  and  ISA.     The  results  in  Table  No.  17  appear  to  be  unusually 

*  From  Bulletin  No.  20,  Eng.  Exp.  Sta.,  University  of  Illinois. 


188 


TESTS    OF    BEAMS    AND    COLUMNS. 


[Cn.  IV. 


low.  Tests  made  at  the  University  of  Wisconsin  in  1908 
indicate  that  with  careful  workmanship  and  testing  an  average 
value  of  about  2000  lbs/in2  can  be  obtained  in  60  days  on 
1:2:4  concrete,  the  results  obtained  being  very  uniform. 

114.  Tests  of  Columns  with  Longitudinal  Reinforce- 
ment only. — The  results  of  a  valuable  series  of  experiments 
made  at  the  Massachusetts  Institute  of  Technology  are  given 
in  Table  No.  15.*  The  concrete  was  1:3:6  broken  stone 
concrete;  the  rods  were  partly  plain  square  rods  and  partly 
twisted  rods,  the  strength  of  the  plain  rods  being  56,000- 
60,000  lbs/in2,  and  of  the  twisted  rods  about  80,000  lbs/in2. 
Where  single  rods  were  used  they  were  placed  in  the  centre, 
and  where  four  rods  were  used  they  were  placed  in  the  form 
of  a  square  one-half  the  dimensions  of  the  column.  The 
columns  were  approximately  thirty  days  old. 

TABLE  No.  15. 
TESTS  OF  REINFORCED   COLUMNS. 

MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY. 


Num- 
ber. 

Cross- 
section. 

Ratio: 
Length 
Diam. 

Number 
of  Rods 
and  Size 
(Square). 

Plain 
or 
Twisted. 

Area  of 
Steel, 
Sq.  In. 

Percent- 
age of 
Rein- 
force- 
ment. 

Crushing 
Strength. 
Lbs/in2. 

1 

8"X8" 

25.5 

1      " 

P 

1 

1.56 

1670 

2 

25.5 

1      " 

T 

1 

1.56 

1985 

3 

18.0 

1      " 

P 

1 

1.56 

1560 

4 

18.0 

1      " 

T 

1 

1.56 

1970 

5 

9.0 

1      " 

P 

1 

1.56 

2160 

6 

9.0 

1      " 

T 

1 

1.56 

2080 

7 

25.5 

1    11" 

P 

1.56 

2.44    . 

2125 

8 

25.5 

1    11" 

T 

1.56 

2.44 

2410 

9 

25.5 

4  f" 

P 

2.25 

3.51 

2840 

10 

25.5 

4  I" 

T 

2.25 

3.51 

2610 

11 

18.0 

4  I" 

T 

2.25 

3.51 

2300 

12 

18.0 

4  f  " 

P 

2.25 

3.51 

2390 

13 

9.0 

4  1" 

T 

4.0 

6.25 

2470 

14 

9.0 

4  i" 

P 

4.0 

6.25 

3810 

15 

10"X10" 

20.4 

1   1" 

P 

1 

1 

2150 

16 

7.2 

1   V 

P 

1 

1 

2000 

17 

7.2 

1   1" 

T 

1 

1 

2284 

18 

14.4 

1  11" 

T 

1.56 

1.56 

2620 

19 

14.4 

1   11" 

P 

1.56 

1.56 

2570 

20 

14.4 

4   a» 

T 

2.25 

2.25 

3000 

21 

14.4 

4  |» 

P 

2.25 

2.25 

2740 

*  Trans.  Am.  Soc.  C.  E.,  Vol.  L,  1903,  p.  487. 


§  H4.] 


TESTS    OF    COLUMNS. 


189 


Grouping  these   tests  in   accordance   with  the   amount  of 
reinforcement  we  have  the  following  average  values : 


Calculated 

Per  Cent 

Average  Strength, 

Strength, 

Reinforcement  . 

Lbs/in2. 

/  =  1470(1  +  19p). 

Lbs/in2. 

f 

1.56 

1904 

1904 

8"X8"  columns. 

2.44 

2267 

2170 

Average  length=  12.4  ft.  | 

3.51 
6.25 

2535* 
3140 

2450 
3250 

/=1800(J  +I9p). 

10"X10"  columns.             { 
Average  length  =11.0  ft.  1 

1.0 
1.56 
2.25 

2145 
2452 
2870 

2145 
2320 
2600 

It  is  evident  that  the  larger  columns  are,  for  like  reinforce- 
ment, stronger  than  the  smaller  columns,  showing  an  effect 
either  of  ratio  of  length  to  diameter  or  of  diameter  directly. 
Little  difference  is  observed  between  plain  and  twisted  4}ars. 
The  effect  of  amount  of  reinforcement  can  be  observed  by  con- 
sidering each  size  separately.  The  results  have  been  studied 
on  the  basis  of  the  theoretical  formula  of  Art.  95,  Chapter  III, 


Pr 


(1) 


in  which  P'/P  represents  the  ratio  of  the  strength  of  the  rein- 
forced to  that  of  the  plain  concrete  column. 

No  results  are  given  for  plain  concrete  columns,  but  as- 
suming that  the  column  with  the  lowest  percentage  of  steel 
follows  the  theoretical  law  the  strength  of  the  ideal  plain  con- 
crete column  is  calculated  to  be  1470  lbs/in2  for  the  first 
group  and  1800  lbs/in2  for  the  second  group,  making  ft  =  20. 
Taking  these  values  then  as  a  basis  the  results  are  plotted  in 
Fig.  54.  Abscissas  represent  per  cent  of  reinforcement  and 
ordinates  the  relative  strengths,  that  of  the  ideal  plain  con- 
crete being  100.  The  theoretical  relation  is  shown  by  the 
straight  line  drawn  for  ft  =  20.  This  value  of  n  corresponds  to 
a  value  of  Ec  of  1,500,000,  which  would  be  a  reasonable  value 


190 


TESTS    OF    BEAMS    AND    COLUMNS. 


[Cn.  IV 


at  rupture  on  the  basis  of  total  deformation,  as  explained  in 
Art.  24.  While  the  results  are  not  sufficiently  numerous  to  be 
at  all  conclusive,  they  do  indicate  that  the  relative  strength 


of  Strength  to  Plain  Co.nccete 

j-  r  to  M  fS  R 

*  §  S  8  8  £  g 

^S 

.^ 

^ 

.^ 

^". 

^^ 

s>^ 

> 

r^ 

.^ 

t   ^s 

s^ 

^ 

'**' 

• 

^ 

^ 

o  «..«» 

^^ 

^^ 

•  8" 

K  8  "Col 

urnns 

1.00 

^^\ 

010 

xlO' 

" 

^s^ 

•z 

)                      1                      2 

>                     v4                      5                      6                     7 

Percentage  of  Reinforcement 


FIG.  54.  —  Tests  of  Reinforced  Columns.     (Mass.  Inst.  of  Technology.) 

of  such  columns  is  fairly  represented  by  the  theoretical  law. 
Calculated  values  corresponding  to  the  theoretical  lines  of  the 
diagram  are  given  by  the  formulas 


and 


/  =  1470(1 

/  =  1800(1 +  19p). 

These  values  are  given  in  the  table  on  p.  153.  Eliminating  the 
longest  columns  of  the  first  group  a  fairly  correct  value  for  the 
ultimate  strength  of  all  would  be  given  by  /  =  1600(l-t-19p) 
(n  is  assumed  equal  to  20). 

The  following  table  gives  results  of  tests  made  at  the  Water- 
town  Arsenal  on  concrete  columns  reinforced  with  longitudinal 
bars  only.  All  columns  were  8  ft.  long  and  approximately 
12"X12"  square;  age,  3J  to  8  months. 


§  H4-] 


TESTS    OF    COLUMNS. 


191 


TABLE  No.  16. 

TESTS  OF  REINFORCED  COLUMNS. 
WATERTOWN  ARSENAL,    1904-1905. 


Reinforcement. 

Com- 

Strength 
of  Plain 

Ratio  of 
Strength 
of  Rein- 

Kind of  Concrete. 

Description. 

Per  Cent. 

pressive 
Strength, 
Lbs/in2. 

Concrete. 
(See  Table 
No.  14.) 

forced 
Concrete 
to  Plain 

Concrete. 

1  •  2  mortar  

8  I"  bars 

2.85 

4200 

3070 

1.37 

1-3       "      

if          ft 

2.87 

3841 

2377 

1.61 

1:4       "      

1  1           «  ( 

2.86 

3377 

1518 

2.22 

1:5       "      ... 

I  (           (  f 

2.86 

2813 

1060 

2.65 

1-5       " 

13  I"    " 

4.63 

3905 

106G 

3  68 

1:1:2  (pebbles)   .  . 

4  f"  twisted 

1.46 

2890 

1720 

.68 

1:2:4 

(  t          i  ( 

1.43 

1990] 

.17 

<               t 

4  f  "  Thacher 

1.03 

1990  | 

.17 

t              ( 

4  f  "  corrugated 

.97 

21801 

.28 

i              i 

4  f  "  twisted 

1.45 

1820  \ 

1710 

.06 

<              t 

8  r    " 

2.86 

3160  | 

1.84 

t              t 

8  £"  Thacher 

2.09 

2760  i 

1.62 

<               t 

8  |"  corrugated 

.94 

2830  J 

1.66 

1:3:6          '          !  ' 

4  |"  twisted 

.44 

1370 

462 

2.96 

1:3:6  (trap-rock).  . 

8  |"  corrugated 

.94 
.93 

2290 
2650 

}  1350 

1.82 

1:2:4  (cinders)  

4  y  '  twisted 

.45 

2095 

871 

2.40 

1:3:6 

4  f  "  bars 

.42 

1932 

(    T  A£5A 

1.82 

*  '      .... 

S  f"    " 

2.83 

3100 

r  lUoU 

; 

2.92 

On  Fig.  55  are  plotted  the  results  of  the  mortar  tests  and  the 
1:2:4  concrete  in  the  same  manner  as  the  values  in  Fig.  54, 
using  as  a  standard  the  results  on  plain  concrete  given  in  Table 
No.  14.  Average  values  have  been  plotted  for  the  columns 
with  percentages  of  .97  and  1.03  and  of  1.43  and  1.45.  Lines 
have  also  been  drawn  representing  the  theoretical  relations  for 
different  values  of  n.  In  the  mortar  tests  the  results  show 
that  for  the  poorer  mortars  the  relative  effect  of  the  steel  is 
high,  corresponding  to  what  would  be  obtained  theoretically 
by  using  a  value  of  ft  =  40  to  50.  In  the  1:2:4  concretes  the 
results  do  not  vary  widely  from  the  theoretical  results  for 
n  =  30,  or  a  value  of  Ec  at  rupture  of  1,000,000. 


192 


TESTS    OF    BEAMS    AND    COLUMNS. 


[Cn.  IV. 


It  is  assumed  in  the  theoretical  discussion  that  the  steel  is 
not  stressed  beyond  its  elastic  limit.  It  is  to  be  noted  that 
in  these  tests  the  stress  on  the  steel  bars  must  have  been  as 
high  as  45,000  to  50,000  lbs/in2,  showing  the  usefulness  of  a 
fairly  high  elastic-limit  steel  in  this  case.  (See  further  dis- 
cussion in  Chapter  V.) 


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ncrete 

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)                  1                   2                   3                   4                   5                   G 

Percentage  of  Reinf orcement 
FIG.  55.— Tests  of  Reinforced  Columns.     (Watertown  Arsenal.) 

Table  No.  17  contains  the  results  of  tests  made  by  Profes- 
sor A.  N.  Talbot  at  the  University  of  Illinois.  The  columns 
were  made  of  1:2:3-3/4  concrete  and  plain  steel  of  39,800 
pounds  per  square  inch  elastic  limit.  The  age  was  from  59  to 
71  days.  Comparing  the  reinforced  with  the  plain  concrete, 


TESTS    OF    COLUMNS. 


193 


TABLE  No.  17. 

TESTS  OF  REINFORCED  COLUMNS. 
UNIVERSITY  OF  ILLINOIS,  1906.* 


No. 

Length. 

Cross-section. 

Reinforcement. 

Crushing  Strength. 
Pounds  per  sq.  in. 

Kind. 

Per  cent. 

Individual 
Test. 

Average 
of  Group. 

1 

4  f-'n.  rods 

1.20 

1587 

] 

3 

7 

12  ft. 

12"X12" 

f    4  f-in.  rods 
\  12   4  in.  ties 
4  f  in.  rods 

1.21 

1862 
1859 

1809 

11 

J    4  f-in.  rods 
\  12    ;-in.  ties 

j  1.21 

1936 

2 

12ft. 

^ 

4  f-in.  rods 

1.52 

1577 

6 

" 

4  f-in.  rods 

1.52 

1600 

10 

" 

J"    4  f-in.  rods 
{  12  |-in.  ties 

}  1.50 

1280 

12 

9ft. 

-    9"X9" 

j     4  f-in.  rods 
9  i-in.  ties 

}l.48 

2335 

1710 

14 

12  ft. 

4  f-in.  rods 
12  t-in.  ties 

|  1.50 

1367 

16 

9ft. 

4  f-in.  rods 
9  1-in.  ties 

|  1.49 

1607 

17 

6ft. 

4  f-in.  rods 

1.47 

220G 

5 

12ft. 

12"X12" 

] 

1710 

8 

n 

9"X9" 

2004 

9 
13 

\t 

12"X12' 

it         << 

I  Plain 

0 

1610 
1709 

>1550 

15 

6  ft. 

t(         it 

. 

1189 

18 

9"  X  9" 

1079 

the  average  strength  of  the  12"xl2"  columns  with  1.2  per 
cent  reinforcement  is  about  1.17  times  as  great,  and  the  9"X9" 
columns  with  1.5  per  cent  reinforcement  is  about  1.10  times 
as  great.  These  tests  indicate  a  less  effect  of  reinforcement 
than  some  of  the  other  tests  quoted.  The  smaller  cross-section 
of  the  columns  containing  the  larger  amount  of  reinforcement 
may  have  been  the  cause  of  the  lower  strength  of  this  group. 
It  is  important  to  note  the  wide  variation  in  the  individual 
results  of  these  and  other  tests;  they  indicate  what  may  be 
expected  in  practice,  and  show  clearly  the  necessity  of  adopt- 
ing conservative  values  of  working  stress.  Careful  measurement 


*  Bulletin  No.  10,  Engineering  Exp.  Sta.,  19C7. 


194  TESTS   OF   BEAMS    AND    COLUMNS.  [Cn.  IV. 

of  distortions  showed  that  tbe  ratio  of  stress  in  steel  to  stress 
in  concrete  varied  from  about  14  at  the  beginning  to  about  27 
at  rupture,  taking  average  values.  The  low  values  for  ultimate 
strength  of  the  reinforced  columns  appeared  to  be  due  to  a 
lower  actual  crushing  strength  of  the  concrete  in  these  columns 
than  in  the  plain  columns. 

115.  Tests  of  Hooped  Columns. — If  a  compression  mem- 
ber be  reinforced  by  bands  or  hoops  closely  spaced,  such  rein- 
forcement will  raise  the  ultimate  strength  by  preventing  lateral 
expansion  under  the  compressive  forces.  It  was  shown  in 
Art.  96  that  under  this  system  of  reinforcement  the  steel 
cannot  be  stressed  to  any  considerable  extent  under  loads 
below  the  usual  elastic  limit  strength  of  the  concrete.  This 
limit  being  exceeded,  however,  the  banding  becomes  very 
effective  in  holding  the  concrete  together  so  that  it  will  endure 
large  deformations  without  rupture,  thus  increasing  greatly 
its  ultimate  strength.  Longitudinal  reinforcement  is  also 
used  with  hoops  or  bands.  This  part  of  the  reinforcement 
will  receive  stress  in  proportion  to  the  longitudinal  deforma- 
tions and  will  thus  be  more  effective  at  low  loads  than  the 
bands.  Results  on  both  forms  of  columns  are  here  given. 

In  1902  and  1903  Considere  *  published  certain  tests  made 
on  columns  reinforced  by  spirally  wound  wire  and  by  longi- 
tudinal rods  or  wire.  His  most  important  results  were  those 
obtained  upon  a  number  of  octagonal  columns  5.9  in.  short 
diameter.  As  a  result  of  these  and  other  tests,  as  well  as  from 
a  theoretical  basis,  he  came  to  the  conclusion  that  steel  in  the 
form  of  spiral  reinforcement  was  2.4  times  as  efficient  as  in  the 
form  of  longitudinal  reinforcement,  presuming  the  spacing  of 
the  wire  to  be  not  great  (}  to  ^  of  the  diameter  of  the  spiral) 
and  that  ordinary  mild  steel  be  used.  It  was  found  also  de- 
sirable to  use  a  small  amount  of  steel  in  the  form  of  longitudinal 
reinforcement.  Tests  on  the  elastic  properties  showed  con- 
siderable deformation  and  set,  but  after  the  first  application  of 

*G&ue  Civil,  1902. 


§  115.] 


TESTS    OF  HOOPED  COLUMNS. 


195 


a  load  the  column  is  relatively  rigid,  with  greatly  increased 
•value  of  E. 

Table  No.  17 A  gives  the  results  of  an  important  series  of 
tests  on  hooped  columns  made  by  Professor  Talbot,  in  1907. 

TABLE  No.  17A. 
TESTS  OF  HOOPED  COLUMNS. 
UNIVERSITY  OF  ILLINOIS,  1907.* 
Concrete  1:2:4;  age,  from  56  to  69  days;  length,  10  ft.;  diam.,  12  ins. 


Group. 

Col.' 
No. 

Reinforcement. 

Cru-hing 
Strength, 
Lbs/in.2 

Kind. 

Size  and  Spacing. 

Per 

Cent. 

Indi- 
vidual 
Tests. 

Aver- 
age of 
Group. 

f 

131 

}                                              \ 

1.08 

2384 

] 

1 

132 

\  No.  16,  2  in.  c.-c.     \ 

1.08 

2150 

j-2239 

I 

133 

J                                     I 

1.05 

2182 

J 

f 

136 

1                                     f 

2.08 

2860 

1 

2 

137 

\  No.  12,  2  in.  c.-c.     \ 

2.07 

2660 

2877 

I 

138 

Electrically 

J                                   1 

2.12 

3110 

J 

welded 

'{ 

146 
147 
148 

bands. 

[•  No.  8,  2  in.  c.-c.       J 

3.22 
3.20 
3.20 

3000 
3715 
2890 

j-3202 

4 

143 

No.  12,  3  in.  c.-c. 

1.39 

2735 

2735 

•{ 

141 
142 

}  No.  12,  4  in.  c.-c.     | 

1.02 
1.02 

2275 
2178 

1  2226 

6( 

171 
172 

High  carbon 

j  No.  7                        { 

0.85 
0.85 

2503 
2506 

}2505 

181 

wire  spiral. 

1                                  r 

1.73 

2718 

] 

182 

[  J  in.                          \ 

1.67 

3800 

13437 

183 

• 

J                                    I 

1.68 

3793 

J 

r 

176 

}                        r 

0.84 

2080 

} 

g    J 

177 

[NO.  7 

0.85 

2203 

\  2168 

1^ 

178 

Mild  steel 

J                        1 

0.84 

2220 

J 

wire  spiral 

f 

186 

i                        r 

1.64 

2068 

1 

9 

187 

Um. 

1.71 

3404 

y  2736 

1 

188 

J                   I 

1.61 

J 

*  Bull.  No.  20,  Eng.  Exp.  Sta.,  Univ  of  111. 


196 


TESTS    OF    BEAMS    AND    COLUMNS. 


[Cn. IV. 


Two  forms  of  hooping  were  used,  electrically  welded  bands 
1  in.  wide  and  of  various  gage  thickness,  and  spirally  wound 
wire  at  a  pitch  of  1  in.  The  steel  used  in  the  bands  had  a 
yield  point  of  about  48,000  Ibs/in2.  The  wire  was  of  two 
kinds,  high  carbon  and  mild  steel.  The  former  had  a  yield 
point  of  115,000  Ibs/in2  for  the  J-m.  size  and  60,000  Ibs/in2 
for  the  No.  7;  the  latter  had  yield  points  for  the  same  sizes 
of  54,000  and  38,500  Ibs/in2,  respectively.  The  columns  were 
10  ft.  long  by  12  in.  in  diameter.  A  thin  film  of  mortar 
covered  the  hooping. 

In  a  study  of  these  tests  it  is  desirable  to  keep  in  mind 
the  two  more  or  less  independent  elements,  namely,  the  ulti- 
mate strength,  and  the  behavior  of  the  specimen  previous  to 
rupture,  which  is  shown  best  by  the  stress  deformation  diagram. 
The  latter  element  may  be  of  greater  importance  than  the 
former. 

As  to  ultimate  strength,  the  results  may  be  compared  by 
groups  with  those  for  plain  concrete  given  in  Table  No.  14A, 
group  2.  The  figures  are  here  brought  together: 


COMPARISON   OF  HOOPED  AND   PLAIN   CONCRETE   COLUMNS. 


Excess  over  Plain 

Group. 

Kind  of  Reinforcement. 

Average 
Amount  of 
Reinforce- 

Average 
Ultimate 
Strength, 

Concrete,  Lbs/in2. 

Per  Cent. 

Lbs/in2. 

Total. 

Per  1% 
Reinforce- 

ment. 

2 

Plain  concrete 

0 

1740 

.... 

.... 

1 

1.07 

2239 

599 

560 

2 

2.09 

2877 

1137 

540 

3 

Bands 

3.21 

3202 

1462 

450 

4 

1.39 

2735 

995 

710 

5 

1.02 

2226 

486 

480 

6 

7 

|  High  carbon  wire     / 

0.83 
1.69 

2505 
3437 

765 
1697 

920 
1000 

8 
9 

|  Mild  steel  wire 

0.84 
1.65 

2168 
2736 

428 
996 

510 
600 

§  115.]  TESTS   OF    HOOPED    COLUMNS.  197 

The  average  increase  per  1%  of  steel  for  the  banded  columns 
is  about  570  lbs/in2;  for  the  high  carbon  wire,  960  lbs/in2: 
and  for  the  mild  steel  wire,  555  lbs/in2.  As  shown  in  Art.  114, 
the  effect  of  longitudinal  reinforcement  may  be  taken,  in 
accordance  with  theory,  as  equal  to  fc(n  — 1)1/100  for  each  1% 
of  steel.  For  the  plain  concrete  columns  in  these  tests  the 
value  of  n  at  rupture  was  about  17,  hence  for  1%  longituinal 
reinforcement  the  strength  should  be  equal  to  1740  X  (17—1)1/100 
or  about  280  Ibs/iri2.  Comparing  this  with  the  results  in  the 
table  it  is  seen  that  the  1%  of  steel  in  the  form  of  bands  added 
about  twice  as  much  to  the  strength  of  the  column,  and  in 
the  form  of  spirally  wound  wire,  from  twice  to  three  and  one-half 
times  as  much.  Furthermore,  it  would  appear  that  the  increase 
in  strength  within  the  limits  of  the  tests  is  about  proportional 
to  the  amount  of  steel  used.  It  should  be  said  that  in  Professor 
Talbot's  analysis  the  strength  of  the  plain  concrete  is  estimated 
for  each  column  by  a  study  of  its  deformation  and  not  from 
the  test  on  plain  columns.  He  thus  arrives  at  Values  for  plain 
concrete  averaging  about  1600  Ibs/in2,  resulting  in  a  still 
better  showing  for  the  reinforcement. 

The  effect  of  hooping  upon  the  deformation  of  the  columns 
and  their  general  behavior  before  rupture  is  of  perhaps  greater 
importance  than  its  effect  upon  ultimate  strength.  The  results 
in  general  are  in  accordance  with  the  discussion  of  Art.  96. 
For  loads  below  that  corresponding  to  the  ultimate  strength 
of  a  plain  concrete  column  there  is  no  strengthening  effect  of 
the  hooping  apparent,  but  beyond  this  load  the  column  shortens 
rapidly  and  the  hooping  comes  into  action.  The  hooped 
columns  in  fact  seem  to  be  somewhat  less  stiff  at  low  loads 
than  the  plain  concrete,  due,  possibly,  to  the  interference  of 
the  bands  in  the  fabrication.  The  total  deformation  at  rup- 
ture is  very  great,  amounting  to  from  6  to  12  times  that  for 
plain  concrete;  and  at  maximum  load  it  is  about  5  times  as 
great.  Scaling  of  the  exterior  shell  occurred  at  loads  corre- 
sponding to  the  ultimate  strength  of  plain  concrete. 

The  deformation  of  plain  concrete  and  of  hooped  concrete 


198  TESTS    OF    BEAMS   AND    COLUMNS.  [Cn.  IV. 

with  different  amounts  of  reinforcement  is  well  shown  in  Fig. 
55a,  in  which  have  been  reproduced  typical  stress-strain  curves 
for  some  of  the  columns  tested.  These  curves  should  be  care- 
fully studied.  They  bring  out  in  a  striking  manner  the  fact 
that  the  chief  effect  of  hooping  is  to  increase  the  "toughness" 
or  " ductility"  of  the  concrete,  to  increase  its  ultimate  strength, 
though  to  a  less  extent,  but  to  cause  little  or  no  change  in  the 
deformation  of  the  column  in  the  early  part  of  the  test.  It 
is  seen  that  up  to  a  stress  of  1200  to  1500  lbs/in2  the  deforma- 
tions are  about  the  same  in  all  columns,  but  that  beyond  this 
the  deformation  of  the  hooped  columns  rapidly  increase.  As 
usually  interpreted,  it  would  appear  that  the  elastic  limit  of 
all  the  columns  is  in  this  region  of  stress  and  is  about  the  same 
for  all.  Doubtless  a  small  set  would  occur  at  still  smaller 
loads,  but  up  to  1200-1500  lbs/in2  the  set  could  not  be  great. 
That  the  hooped  column  would  act  in  this  respect  much  like 
mild  steel  is  indicated  by  the  test  on  column  No.  173  in  which 
the  load  was  removed  and  reapplied  with  results  shown  in  the 
figure. 

Tests  made  at  the  Watertown  Arsenal  in  1905  *  showed 
results  very  similar  to  those  quoted  above.  The  reinforcement 
consisted  of  riveted  bands  1.5x0.12  in.  and  longitudinal 
angle  bars  1X1  Xj  in.  The  columns  were  1:2:4  concrete 
5  and  6  months  old  and  were  10|  in.  in  diameter  by  8  ft.  long. 
The  entire  column  was  inclosed  by  the  bands.  The  results 
are  given  in  Table  No.  17B. 

The  additional  strength  of  the  hooped  columns  over  the 
-plain  concrete  for  each  1%  of  reinforcement  was  819  lbs/in2 
for  13  hoops,  1120  lbs/in2  for  25  hoops  and  1140  lbs/in2  for 
47  hoops.  The  additional  strength  due  to  the  angle  bars  over 
the  same  column  without  the  bars  was  797  lbs/in2  for  the 
one  with  13  hoops  and  761  lbs/in2  for  that  with  25  hoops. 
These  values  are  about  equal  to  the  highest  obtained  by  Talbot. 

Of  much   significance  in  these   tests  also  is  the    character 

*  Tests  of  Metals,  1906. 


§115.] 


TESTS   OF    HOOPED    COLUMNS. 


199 


•qoui  'I)s  .T9d_spuno<£  issoa^g 


200 


TESTS    OF    BEAMS    AND    COLUMNS. 


[On.  IV. 


•qoui  -bs  jed  sptraoj  tsssj^g 


§  H5.J 


TESTS    OF    HOOPED    COLUMNS. 


201 


TABLE  No.  17u. 

TESTS  OF  HOOPED  COLUMNS. 
WATERTOWN  ARSENAL,  1905. 


Reinforcement. 

Strength, 
Lbs/in2. 

Kind. 

Per  Cent 
Hoops. 

Per  Cent 
Longitudinal. 

Plain  concrete  . 

1.0 
1.0 

1.8 
1.8 
3.4 

'  Yd 

1.0      . 
.0 
1.0 
.0 

1413 
2232 
3029 
3428 
4189 
5289 

13  hoops    

13  hoops  4  L's 

25  hoops  

25  hoops  4  L's    . 

47  hoops  .  .  . 

of  the  stress  deformation  curves,  as  shown  in  Fig.  556.  The 
general  effect  of  banding  is  the  same  as  indicated  in  Talbot's 
tests;  it  increases  greatly  the  ultimate  strength  but  does  not 
stiffen  the  column  up  to  the  ultimate  strength  of  plain  con- 
crete. The  effect  of  longitudinal  reinforcement  on  the  defor- 
mation curve  is  marked.  In  stiffening  the  column  at  early 
stages,  or  in  raising  its  elastic  limit,  it  has  a  much  greater 
effect  than  an  equal  amount  of  band  reinforcement. 

Thus  at  the  deformation  of  .00015,  corresponding  to  a 
stress  in  the  longitudinal  steel  of  45,000  lbs/in2,  1%  of  longi- 
tudinal steel  increases  the  resistance  for  the  column  of  13 
hoops,  from  1520  to  2080,  or  about  560  lbs/in2,  whereas  adding 
.8%  in  hoops  increases  it  to  1780,  or  260  lbs/in2.  Even  47 
hoops,  or  2.4%  of  added  steel,  adds  no  more  strength  at  this 
deformation  than  ]%  of  longitudinal  steel.  If  the  elastic 
limit  of  the  longitudinal  steel  is  45,000  lbs/in2,  then  its  ulti- 
mate resistance  would  be  reached  at  a  deformation  of  .0015, 
and  at  this  deformation  1%  of  such  reinforcement  would, 
theoretically,  add  45,000  X  .01  =  450  lbs/in2.  This  figure  appears 
to  have  been  exceeded  in  these  tests. 

Table  No.  18  gives  results  of  a  series  of  experiments  on 
hooped  columns  conducted  by  Bach.*  The  columns  were  of 

*  Quoted  from  Morsch,  Eisenbetonbau,  p.  70. 


202 


TESTS    OF    BEAMS    AND   COLUMNS. 


IV. 


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5  115.] 


TESTS    OF    HOOPED    COLUMNS. 


203 


octagonal  form  with  short  diameter  equal  to  275  mm.  and  height 
of  1  m.  The  diameter  of  the  spirals  was  245  mm.  The  con- 
crete was  1:4  gravel  concrete  5-6  months  old.  Each  result  is 
the  average  of  three  tests,  except  in  the  case  of  the  unreinforced 
concrete,  where  four  tests  were  made.  The  steel  was  mild  steel. 
The  strength  is  calculated  with  reference  to  the  gross  section 
of  the  column. 

It  is  difficult  to  draw  any  definite  conclusions  from  these 


Strength,  Pounds  per  Square  Inch 

^^^ 

^•^ 

< 

ftp 

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^ 

^^ 

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^ 

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0 

^ 

Percent  Spiral  Reinforcement 
FIG.  56.— Tests  of  Hooped  Columns  (Bach). 

tests  as  to  the  relative  value  of  longitudinal  and  hoop  rein- 
forcement. However,  some  suggestions  of  value  may  be 
obtained  from  a  consideration  of  those  tests  where  the  longi- 
tudinal steel  was  .25  and  1.22%.  These  include  all  of  the 
first  three  groups  and  the  first  of  the  fourth  group.  These 
results  are  plotted  in  Fig.  56  in  two  groups,  corresponding  to 
the  two  percentages  of  longitudinal  steel,  using  the  percentages 
of  spiral  steel  as  abscissae.  Approximate  average  straight  lines 


204  TESTS    OF    BEAMS   AND    COLUMNS.  [Cn.  IV. 

are  drawn  through  each  group.  They  indicate,  roughly,  that  the 
strength  is  increased  by  425  lbs/in2  for  each  percentage  of 
spiral  reinforcement  for  both  groups  and  that  the  strength  of 
the  upper  group  is  about  800  lbs/in2  greater  than  the  lower 
group,  showing  a  strengthening  effect  of  about  800  lbs/in2  for 
1%  additional  longitudinal  reinforcement  when  used  with 
spirals.  Thus,  so  far  as  these  tests  go,  the  longitudinal  steel  is 
fully  as  effective  as  the  spiral  steel  in  the  proportions  here 
used.  Notice  that  the  strength  of  groups  V  and  VI  is  rela- 
tively less  than  the  others.  This  appears  to  be  due  to  the 
wider  spacing  of  the  spirals  in  these  groups.  These  results, 
as  compared  to  those  already  quoted,  are  doubtless  modified 
by  the  fact  that  there  was  a  considerable  thickness  of  con- 
crete outside  of  the  spirals  and  generally  such  outer  shell  will 
crack  and  fall  away  sometime  before  final  failure.  The 
values  given  refer  to  gross-section. 

Some  general  results  of  column  tests  made  by  the  Depart- 
ment of  Buildings  of  Minneapolis  in  1907  are  given  in  Table 
No.  18A.  The  concrete  was  1 : 2 :  3^  mixture  and  generally 

116.  Conclusions  as  to  Strength  of  Columns — From  the 
results  of  tests  quoted  we  may  draw  the  following  conclu- 
sions :  that  the  strength  of  plain  concrete  columns  of  1:2:4 
mixture  at  60  days  may  be  taken  at  from  1600  to  1800  lbs/in2; 
that  very  great  gain  in  strength  is  shown  for  both  plain  and 
reinforced  concrete  by  the  use  of  richer  mixtures;  that  the 
strength  of  columns  reinforced  with  longitudinal  rods  only 
(or  when  fastened  together  at  wide  intervals)  may  be  estimated 
in  accordance  with  theory,  but  that  the  density  and  rigidity 
of  the  concrete  itself  is  apt  to  be  less  in  the  reinforced  than  in 
the  plain  column,  so  that  for  small  percentages  of  longitudinal 
reinforcement  the  gain  in  strength  is  small;  that  hooped 
columns  without  longitudinal  steel  show  greatly  increased 
deformation  before  rupture  and  a  much  higher  ultimate  strength 
than  columns  having  the  same  amount  of  longitudinal  steel 
and  no  hoops,  but  that  such  columns  generally  show  less  stiff- 
ness below  the  elastic  limit  than  the  plain  concrete;  that  the 


§  116a.]  TESTS    OF    HOOPED    COLUMNS.  205 

addition  of  longitudinal  steel  to  hooped  columns  increases 
greatly  the  elastic  limit  of  the  column  and  also  its  ultimate 
strength,  its  effect  upon  the  latter  within  ordinary  limits  being 
about  as  great  as  an  equal  amount  of  hooping  and  generally 
greater  than  the  amount  calculated  on  the  basis  of  the  elastic 
imit  of  the  steel. 

n6a.  Effect  of  Lenglh  of  Column  on  Compressive 
Strength. — Comparing  the  results  on  plain  concrete  columns, 
p.  186,  with  the  tests  on  cubes,  pp.  11-14,  it  is  evident  that 
the  strength  of  the  column  is  materially  less.  While  there  is 
thus  a  very  considerable  reduction  of  strength  as  compared  to 
the  cube,  there  appears  to  be  little  difference  in  the  strength  of 
columns  of  various  lengths  up  to  15  to  20  diameters.  A  series 
of  tests  made  at  the  Watertown  Arsenal  *  for  the  Aberthaw 
Construction  Co.  on  12"  XI 2"  columns  gave  practically  the 
same  results  for  all  lengths  from  2  ft.  to  14  ft.,  the  average  of 
all  being  957  lbs/in2  for  hand-mixed  and  1099  lbs/in2  for 
machine-mixed  concrete.  The  temperatures  were,however,  low, 
and  the  results  are  not  a  fair  criterion  as  to  absolute  strength. 

In  the  tests  of  Table  No.  15  the  difference  in  average  results 
upon  the  8"X8"  columns  and  those  on  the  10"X10"  size  is 
marked.  But  comparing  results  for  each  size  among  them- 
selves there  is  little  or  no  effect  noticeable  up  to  25  diameters. 
Numbers  2  and  3  are  reported  as  having  failed  by  buckling, 
but  these  average  practically  the  same  as  Nos.  1  and  4.  From 
these  tests  it  would  appear  therefore  that  no  account  need  be 
taken  of  length  of  column  below  about  20  diameters,  although 
caution  should  be  used  in  accepting  these  results  as  conclu- 
sive. In  the  case  of  hooped  columns  the  effect  of  buckling 
is  evident  for -shorter  lengths,  inasmuch  as  this  kind  of  column 
has  a  sufficient  toughness  to  permit  of  considerable  deforma- 
tion before  failure. 

Considering  the  comparatively  brittle  nature  of  the  column 
with  longitudinal  reinforcement  only,  its  use  for  columns  of 
slender  proportions  should  be  discouraged.  The  banded  or 
hooped  columns  is  much  more  reliable  for  such  work. 

*  Tests  of  Metals,  1897. 


206 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 


117.  Fatigue  Tests  of  Reinforced  Concrete.— Important 
experiments  conducted  by  Professor  J.  L.  Van  Ornum  *  on 
reinforced  beams  indicate  an  effect  under  repeated  application 
of  loads  similar  to  that  which  he  found  for  mortar  and  concrete 
in  compression  as  mentioned  on  p.  25.  In  the  case  of  beams 
the  failure  under  repeated  loads  appeared  to  be  largely  a  gradual 
fracture  in  diagonal  tension,  ending  with  a  compression  failure. 
The  number  of  repetitions  required  to  produce  failure  varied 


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4000          8000         12000       .16000        20000        24000        28000 
Number  of  Repetitions  necessary  to  produce  failure 

FlG.  57. 


32000 


with  the  load  applied,  rupture  being  ultimately  produced  after 
several  thousand  repetitions  for  loads  as  low  as  55  and  60%  of 
the  usual  ultimate  strength.  The  most  important  of  his  results 
are  indicated  in  Fig.  57,  taken  from  his  paper,  showing  the 
number  of  repetitions  required  to  produce  failure  at  various 
values  of  maximum  load  in  percentage  of  the  usual  ultimate 
load. 

The  change  in  the  modulus  of  elasticity  was  also  investigated, 
and  it  was  found  that  under  repeated  loads  not  ultimately 


*  Trans.  Am.  Soc.  C.  E.,  1907,  LVIII,  p.  294. 


§  117.]  HOOPED   COLUMNS,  207 

causing  rupture  the  concrete  soon  became  perfectly  elastic,  with 
a  value  of  the  modulus  of  about  two-thirds  of  its  initial  value. 
At  loads  ultimately  causing  rupture  the  modulus  became  for 
a  time  nearly  constant,  but  rapidly  decreased  as  rupture  was 
approached. 

These  tests  indicate  that  concrete  when  repeatedly  loaded 
beyond  about  50%  of  its  ordinary  ultimate  strength  will  not 
remain  indefinitely  elastic  and  will  fail.  This  limit  may  be 
called  the  permanent  elastic  or  fatigue  limit  of  concrete.  It  is 
of  much  importance  in  relation  to  working  stresses. 

In  some  repeated  load  tests  at  the  University  of  Pennsyl- 
vania,* loads  were  applied  many  thousands  of  times,  producing 
stresses  in  the  concrete  of  from  25  to  40%  of  its  ultimate 
strength  (2460  lbs/in2  in  cube  form).  Some  set  in  the  com- 
pressive  concrete  was  observed  in  all  cases  after  the  first  few 
applications  of  the  load.  At  a  stress  of  25%  of  the  ultimate 
the  set  remained  practically  constant  for  360,000  repetitions; 
at  a  stress  of  32%  of  the  ultimate  the  set  continued  to  increase, 
reaching  a  value,  in  the  case  of  one  beam,  of  about  twice  as 
much  at  500,000  repetitions  as  at  50,000;  and  at  a  stress  of 
40%  of  the  ultimate  the  set  increased  somewhat  more  rapidly. 
It  was  not  found,  however,  that  at  these  stresses  the  ultimate 
strength  of  the  beam  was  affected.  In  these  tests  hair  cracks 
formed  in  the  tension  side  at  loads  producing  stresses  of 
10,000  to  18,000  lbs/in2  in  the  steel.  These  gradually  extended 
well  towards  the  neutral  axis. 

*  Eng.  Record,  Vol.  58,  1908,  p.  90. 


CHAPTER  V. 
WORKING  STRESSES  AND  GENERAL  CONSTRUCTIVE   DETAILS 

ii  8.  Working  Stresses  and  Factors  of  Safety. — In  the  design 
of  steel  structures  it  has  come  to  be  the  practice  to  make  use 
of  definite  working  stresses  rather  than  factors  of  safety.  These 
working  stresses  are  based,  for  the  most  part,  on  the  permanent 
elastic-limit  strength  of  the  material,  although  the  margin  of 
safety  between  the  elastic-limit  and  the  ultimate  strength  (indi- 
cated by  strength  and  ductility)  receives  consideration.  The 
working  stresses  are  made  sufficiently  below  the  elastic  limit  to 
provide  for: 

(a)  Variations  and  imperfections  in  material  and  work- 
manship. 

(&)  Uncalculated   stresses,   such  as  secondary  stresses, 
stresses  due  to  unequal  settlement,  and,  -usually, 
those  due  to  temperature  changes, 
(c)   Dynamic  effect  of  live  load  if  not  provided  for  by  an 

allowance  for  impact. 
J(d)  Possible  increase  in  live  load  over  that  assumed,  or 

rare  applications  of  excessive  loads. 
(e)  Deterioration  of  the  structure. 

The  more  accurately  the  various  elements  are  determined  in 
any  case  the  closer  may  the  working  stress  approach  the 
elastic  limit.  Where  the  dynamic  effect  of  the  live  load  does 
not  enter,  or  is  otherwise  fully  provided  for,  and  where  items 
(d)  and  (e)  are  of  small  moment,  working  stresses  for  steel  struc- 
tures will  vary  from  about  one-half  to  two-thirds  the  elastic- 
limit  strength  of  the  material.  Were  it  absolutely  certain  that 

208 


§  119.]       WORKING  STRESSES  AND  FACTORS  OF  SAFETY.       209 

the  elastic  limit  of  the  material  would  never  be  exceeded  in 
any  emergency,  then  the  margin  of  strength  between  the  elastic 
limit  and  the  ultimate  strength  would  be  of  no  importance. 
This  is,  however,  not  the  case,  and  under  actual  conditions  of 
service  there  is  a  very  considerable  element  of  safety  in  the 
fact  that  the  ultimate  strength  is  in  most  materials  much 
higher  than  the  elastic  limit.  Stated  in  another  way,  a  designer 
would  never  use  a  working  stress  of  one-half  or  two-thirds  the 
elastic  limit  in  a  material  where  the  ultimate  strength  did  not 
considerably  exceed  this  limit.  While  therefore  the  working 
stresses  are  selected  chiefly  with  reference  to  the  elastic  limit, 
the  ultimate  strength  also  receives  consideration. 

In  recent  years  most  designers  base  their  calculations  on 
certain  working  stresses  selected  as  above  indicated.  Formerly, 
and  to  some  extent  now,  calculations  are  based  on  specified 
"factors  of  safety"  referred  to  ultimate  strengths.  In  either 
case  both  the  elastic  limit  and  the  ultimate  strength  must  be 
considered  in  the  design,  and  experienced  designers  will  arrive 
at  about  the  same  results  by  either  method.  In  reinforced-con- 
crete  design  the  problem  is  complicated  by  the  use  of  two  unlike 
materials  whose  elastic  limits  and  ultimate  strengths  are  not 
similarly  related.  Furthermore,  as  the  materials  are  stressed 
beyond  their  elastic  limits  the  stresses  do  not  necessarily  increase 
in  proportion  to  the  load,  so  that  "if  working  stresses  of  one- 
fourth  the  ultimate  are  selected,  for  example,  the  corresponding 
load  may  be  considerably  greater  or  less  than  one-fourth  the 
ultimate  load.  This  condition  makes  it  especially  desirable 
to  consider  ultimate  strength,  and  is  an  argument  for  the  use 
of  the  "factor-of -safety"  method. 

119.  Relative  Effect  of  Dead  and  Live  Loads. — The  ten- 
dency of  practice  in  the  treatment  of  live-load  stresses  is  to  re- 
duce them  to  equivalent  dead-load  stresses  by  the  application 
of  some  sort  of  impact  formula  or  by  other  means  of  estimation. 
The  resulting  stresses  are  then  considered  on  the  same  basis 
as  the  usual  dead-load  stresses  and  a  single  set  of  working  stresses 
applied.  This  method  is  simple,  logical,  and  tends  to  facilitate 


210  WORKING  STRESSES.  [On.  V, 

a  proper  adjustment  of  the  design  to  the  conditions.  Separate 
working  stresses  will  give  equally  satisfactory  results  when 
properly  selected,  but  the  system  is  not  as  flexible  or  convenient 
as  the  method  of  the  single  working  stress  with  impact  coeffi- 
cients. 

The  question  of  impact  coefficients,  or  the  relation  between 
live-  and  -  dead-load  working  stresses,  requires  little  special 
attention  in  connection  with  reinforced  concrete  structures. 
It  is  essentially  the  same  as  it  is  in  the  case  of  steel  structures, 
excepting  as  the  amount  of  impact  may  be  modified  by  the 
structure  itself.  In  steel  railroad  structures  of  short  span,  for 
example,  the  impact,  or  dynamic  effect  of  live  load,  is  usually 
assumed  to  be  about  100%  of  the  live  load  stresses.  Experi- 
ments show  that  this  is  probably  not  too  high  and  that  the 
actual  stresses  from  live  load  may  be  100%  greater  than  the 
static  stresses,  due  largely  to  the  effect  of  unbalanced  locomo- 
tive wheels.  Where  a  large  amount  of  ballast  intervenes  be- 
tween the  load  and  the  structure  the  impact  is  doubtless  much 
less.  In  the  case  of  concrete  structures  the  great  mass  of  the 
concrete  undoubtedly  tends  to  reduce  the  effect  of  impact  and 
vibration,  or  to  localize  such  effect  more  than  in  a  steel  structure. 
The  conditions  involved  in  concrete  designing,  therefore,  are 
likely  to  be  favorable  as  regards  impact  and  may  permit  the 
use  of  lower  coefficients  than  are  used  for  steel  structures.  The 
proper  coefficient  to  use,  or  the  relation  between  live-  and  dead- 
load  working  stresses,  varies  much  under  different  conditions 
and  must  be  left  to  the  judgment  of  the  designer,  or  to  formulas 
or  rules  prepared  especially  for  the  purpose.  Further  discussion 
of  this  question  will  not  be  undertaken  here. 

In  buildings  it  is  the  practice  in  steel  construction  to  use  a 
single  working  stress,  no  account  being  taken  directly  of  any 
special  effect  of  the  live  load.  Allowance  is  made  in  the  design 
of  large  girders  and  columns  which  receive  their  load  from 
large  areas  for  the  fact  that  such  large  areas,  especially  if  on  two 
or  more  floors,  are  seldom  or  never  loaded  to  the  extent  assumed 
for  smaller  areas.  This  allowance  varies  with  different  conditions, 


§  120.]  SAFE  WORKING  STRESSES.  211 

but  relates  solely  to  the  selection  of  the  amount  of  live  load 
rather  than  to  its  effect.  In  a  building,  when  heavily  loaded 
with  its  live  load,  the  portion  of  the  load  which  is  in  motion 
and  capable  of  producing  a  dynamic  effect  is  generally  but  a 
very  small  percentage  of  the  total  live  load.  In  most  cases, 
therefore,  in  building  construction  it  is  not  necessary  to  treat 
the  live-load  stresses  differently  from  the  dead-load  stresses, 
and  the  design  is  based  on  a  single  set  of  working  stresses. 
Special  cases  will  arise,  however,  where  the  dynamic  effect  of 
the  live  load  requires  consideration,  as,  for  example,  in  the  case 
of  floors  supporting  moving  machinery. 

Whatever  the  effect  of  live  load  may  be  it  can  more  readily  be 
taken  account  of  by  adding  to  the  resulting  live-load  stresses 
a  percentage  which,  in  the  judgment  of  the  engineer,  will  reduce 
them  to  their  dead-load  equivalent,  and  then  apply  a  single 
set  of  working  stresses,  or  factor  of  safety,  to  the  sum  of  the 
stresses.  The  discussion  of  working  stresses  in  the  following 
articles  will  relate  to  the  proper  basal  working  stress  for  dead 
load,  or  for  live  load  suitably  increased  for  impact. 

BEAMS. 

120.  Working  Formulas.— From  the  analysis  and  results  of 
experiments  discussed  in  preceding  chapters  there  would  appear 
to  be  no  good  reason  why  the  rational  formulas  as  developed  in 
Chapter  III  should  not  be  used  in  designing.  No  empirical 
formula  is  needed .  Furthermore,  in  the  judgment  of  the  authors, 
the  simple  formulas  based  on  the  straight- line  stress  variation 
should  be  used  for  purposes  of  design,  safe  working  stresses 
being  employed.  These  formulas  are  practically  correct  for 
such  working  stresses,  aiid  there  seems  to  be  no  more  reason 
to  use  formulas  designed  only  to  express  ultimate  strength  than 
there  is  in  the  case  of  .wooden  or  cast-iron  beams  where  the 
conditions  are  similar.  It  is,  however,  desirable  that  the  work- 
ing stresses  be  selected  with  some  reference  to  ultimate  strength, 
although  with  principal  reference  to  elastic  strength. 


212  WORKING  STRESSES.  [Cn.  V. 

121.  Working  Stresses  in  Concrete  and  Steel.— The  strength 
of  a  beam  is  limited  usually  by : 

(a)  The  compressive  strength  of  the  concrete, 

(b)  The  elastic-limit  strength  of  the  steel,  or 

(c)  The  strength  of  the  beam  in  diagonal  tension. 

In  this  article  the  first  two  elements  only  will  be  considered. 

From  tests  relative  to  elastic  limit,  such  as  those  of  Bach 
and  Van  Ornum  (see  Chapters  II  and  IV j,  it  would  appear  that 
the  permanent  elastic  limit  of  concrete  is  from  50%  to  60% 
of  its  ultimate  strength  as  determined  in  the  usual  man- 
ner. If  a  factor  of  safety  of  two  be  applied  to  the  elastic -limit 
strength  to  provide  for  items  (a),  (6),  and  (c)  of  Art.  118,  we  will 
have  a  dead-load  basal  working  stress  of  25%  to  30%  of  the 
ultimate  strength  as  determined  by  tests  on  cubes.  Taking  this 
ratio  at  30%,  the  data  of  Chapter  II  show  that  the  working 
stresses  for  concrete  of  the  usual  proportions  (1:2:4  to  1:2J:5) 
should  range  from  about  550  to  650  lbs/in2.  A  value  of  600 
lbs/in2  is  commonly  used  and  should  imply  a  strength  of  about 
2000  lbs/in2  in  cube  form  in  60  days.  As  explained  further  on, 
the  stress  in  the  concrete  does  not  increase  in  proportion  to  the 
load  on  the  beam,  so  that  a  working  stress  of  30%  of  the  com- 
pressive strength  will  give  a  factor  of  safety  against  failure  of 
about  4.  It  is  to  be  noted  also  that  the  strength  of  the  con- 
crete increases  with  age.  On  the  whole,  therefore,  a  stress  of 
30%  of  the  strength  at  60  days  may  be  considered  as  conserva- 
tive practice. 

With  respect  to  the  steel  it  is  to  be  observed  that  its  elastic 
limit,  or  more  correctly  speaking,  its  yield-point,  determines 
not  only  the  elastic  limit  strength  of  the  beam,  but  also,  approx- 
imately, its  ultimate  strength,  and  the  working  stress  should 
be  selected  with  this  in  view.  If,  for  example,  the  working 
stress  is  taken  at  one-half  the  elastic  limit  strength  of  the  steel, 
the  factor  of  safety  will  be  two  as  to  elastic  strength  and 
slightly' more  than  this  as  to  ultimate  strength,  whereas  with 
respect  to  the  concrete  the  factor  of  safety  will  be  more  than  four. 

It  is  desirable  to  study  somewhat  more   closely  the  varia- 


§  121.] 


SAFE   WORKING    STRESSES. 


213 


tion  in  the  stress  in  steel  and  outer  concrete  fibre  in  a  beam 
subjected  to  an  increasing  load.  Assume  a  concrete  having  an 
ultimate  compressive  strength  of  2000  lbs/in2,  and  let  working 
stresses  be  assumed  of  500  lbs/in2  in  the  concrete  and  15,000 
lbs/in2  in  the  steel.  Suppose  the  beam  loaded  so  as  to  cause 
these  respective  stresses.  Represent  this  load  on  the  axis  OX, 
Fig.  57a,  by  Oa  and  erect  ordinates  ab  and  ac  representing  to 
some  scale  the  stresses  of  500  and 
15,000  lbs/in2,  respectively.  Let  the 
load  now  be  doubled  and  represented 
by  the  abscissa  Oa' ',  equal  to  twice 
Oa.  The  stress  on  extreme  fibre 
must  now  be  calculated  with  reference 
to  the  true  stress-strain  diagram  of 
the  concrete,  which  will  be  assumed 
a  parabola.  Following  the  method 
of  Art.  65  we  find  the  stress  in  the 
concrete  to  be  950  lbs/in2;  the  stress 
in  the  steel  will  be  30,000  lbs/in2. 
Proceeding  in  this  way  with  increased 
loads,  we  finally  reach  the  ultimate 
strength  of  the  concrete  of  2000 
lbs/in2.  The  corresponding  stress 

in  the  steel  will  be  89,400  lbs/in2.  The  ordinates  of  the 
curve  OAB  represent  the  progressive  increase  in  outer  fibre 
stress  and  those  of  the  curve  0(7  represents  th^t  in  the 
steel,  assuming  for  this  purpose  a  steel  whose  elastic  limie 
is  not  less  than  the  maximum  stress.  This  diagram  shows 
clearly  the  relative  change  in  stress  in  concrete  and  steel  under 
increasing  loads.  It  shows  that  the  ultimate  load,  as  fixed  by 
the  ultimate  strength  of  the  concrete,  is  about  5.75  times  the 
load  which  produces  the  working  stress  of  500  lbs/in2.  The 
stress  in  the  steel  increases  nearly  in  proportion  to  the  load  and 
the  stress  corresponding  to  the  ultimate  strength  of  concrete  is 
about  89,400  lbs/in2,  or  about  six  times  the  working  stress  of 
15,000  lbs/in2. 


2000 


1500 


1000 


500 


£0,000 


40,000 


60,000 


80,000 

Y 

100,000 


/ 

< 

^b 

j/ 

V? 

/ 

/ 

6 
a 

a' 

x 

\ 

\. 

\ 

\ 

••*-? 

\ 

\ 

\ 

^ 

FIG.  57a. 


214  WORKING    STRESSES.  [Cn.  V. 

Returning  now  to  the  question  of  working  stresses  in  the 
steel,  it  is  seen  that  if  a  stress  is  to  be  used  so  as  to  utilize  fully 
the  ultimate  strength  of  the  concrete,  such  stress  cannot  exceed 
one-sixth  of  the  elastic  limit  of  the  steel.  This  would  give  a 
factor  of  safety  of  six  as  to  ultimate  strength  of  the  beam, 
which  is  a  much  larger  factor  than  necessary,  especially  as 
regards  such  a  material  as  steel.  On  the  other  hand,  suppose 
the  working  stress  in  the  steel  be  selected  at  one-half  its  elastic 
limit,  assumed  in  this  case  to  be  30,000  lbs/in2.  Under  in- 
creasing loads  the  beam  will  reach  its  elastic  limit  as  to  both 
concrete  and  steel  at  about  double  its  working  load,  but  as  to 
ultimate  strength  there  is  still  a  large  margin  (about  60%) 
with  respect  to  the  concrete,  but  only  a  very  small  margin 
with  respect  to  the  steel.  If  it  is  desirable  to  utilize,  for  emer- 
gency purposes,  a  larger  part  of  the  ultimate  compressive 
strength  of  the  concrete,  the  working  stress  in  the  steel  must 
therefore  be  selected  so  as  to  give  the  desired  margin  of  strength 
without  much  exceeding  its  elastic  limit.  Considering  the  fact 
that  in  well-designed  beams  the  steel  stress  at  failure  will  con- 
siderably exceed  its  elastic  limit,  a  working  stress  of  one-half 
the  elastic  limit  will  give  a  factor  of  safety  against  ultimate 
failure  of  about  2J,  and  a  working  stress  of  one-third  will  give 
a  factor  of  3J  to  4.  Under  ordinary  conditions  a  working 
stress  of  about  four-tenths  of  the  elastic  limit,  considered  as 
the  yield-point  of  the  material,  would  appear  to  be  desirable. 
With  a  working  stress  in  the  concrete  of  one-half  its  elastic 
limit  the  beam  will  then  have  a  factor  of  safety  as  regards  elastic 
limit  of  about  two  (determined  by  the  concrete),  and  as  regards 
ultimate  strength  its  factor  of  safety  will  be  at  least  five  relative 
to  the  concrete  and  about  three  relative  to  the  steel.  Its 
elastic  limit  is  thus  determined  by  the  concrete  and  its  ultimate 
strength  by  the  steel,  which  may  be  considered  as  satisfactory 
conditions.  The  greater  uniformity  and  reliability  of  the  steel, 
as  compared  to  the  concrete,  should  be  noted  in  ths  connection. 

In  determining  the  relative  working  stresses  in  steel  and 
concrete  some  consideration  should  be  given  to  the  question  of 


§122.]  SAFE    WORKING    STRESSES.  215 

repeated  loads.  Where  a  large  percentage  of  the  load  is  live 
load  and  subject  to  frequent  repetitions,  a  relatively  low  work- 
ing stress  in  the  concrete  may  well  be  employed  in  order  that 
elastic  conditions  may  be  maintained.  As  regards  the  steel, 
more  perfect  elasticity  exists  up  to  a  definite  point,  and  hence 
repetition  of  load  need  not  be  considered  in  the  selection  of  its 
working  stresses. 

The  working  stresses  in  the  steel  should  also  be  considered 
with  reference  to  its  distortion.  High  working  stresses  involve 
large  distortions,  and  hence  a  greater  degree  of  incipient  rupture 
in  the  concrete.  This  condition  is  probably  of  little  moment 
in  most  cases  so  fas  as  it  concerns  the  appearance  of  the  con- 
crete, but  experiments,  such  as  noted  in  Art.  117,  show  that 
these  cracks  may  be  of  some  consequence,  and  their  influence 
on  the  possible  corrosion  of  the  steel  is  not  yet  well  determined. 
In  this  connection  it  may  be  noted  that  under  working  condi- 
tions the  actual  stress  in  the  steel  is  generally  less  than  the  cal- 
culated, owing  to  the  tensile  resistance  of  the  concrete.  The 
deformations  will  therefore  also  be  proportionally  less.  The 
deformations  of  the  concrete  are  also  of  importance  with  refer- 
ence to  their  effect  on  diagonal  tensile  stresses,  as  explained  in 
Art.  109.  Low  unit  stresses  in  the  steel  are  greatly  to  be  pre- 
ferred on  this  account.  It  will  also  be  shown  in  Art.  133  that 
very  little  is  to  be  gained  in  economy  by  using  high  stresses. 
Considering  this  fact  and  the  objections  above  mentioned,  it 
would  seem  that  a  stress  of  16,000  lbs/in2  should  be  considered 
about  the  maximum  desirable  value,  irrespective  of  the  quality 
of  the  steel  used.  A  lower  value  is  to  be  recommended.  Finally, 
as  the  result  of  this  analysis,  we  may  conclude  that  the  basal 
working  stress  in  the  steel  should  not  exceed  about  40%  its 
elastic  limit  nor  exceed  16,000  lbs/in2. 

122.  Quality  of  Steel. — As  stated  in  Art.  34,  there  exists 
considerable  difference  of  opinion  as  to  the  quality  of  steel  to 
be  desired,  especially  with  reference  to  the  use  of  soft  or  hard 
material,  or  steel  with  low  or  high  elastic  limits.  Certainly  a 
material  as  hard  as  that  formerly  denominated  "  hard  bridge 


218  WORKING    STRESSES.  [Cn.  V. 

steel "  is  entirely  suitable  for  reinforced  construction.  Such 
material  has  an  elastic  limit  of  about  40,000  lbs/in2.  Much 
material  has  been  used  of  an  elastic  limit  of  45,000  to  50,000 
lbs/in2  and  even  higher,  but  a  value  beyond  this  is  not  to  be 
desired.  Practice  tends  to  the  use  of  one  of  two  grades  of 
material;  the  ordinary  medium  or  mild  steel  having  a  yield- 
point;  in  the  sizes  suually  employed,  of  from  35,000  to  40,000 
lbs/in2,  and  various  special  bars  having  a  high  elastic  limit  of 
50,000  lbs/in2  or  more.  The  medium  steel  will  permit,  under 
ordinary  conditions,  a  working  stress  of  14,000  to  16,000  lbs/in2, 
and  a  steel  of  higher  elastic  limit  is  of  doubtful  wisdom  unless 
a  high  factor  of  safety  is  desired.  The  ductility  of  the  high 
elastic  limit  material  of  the  usual  quality  is  often  not  as  great 
as  desirable.  As  regards  ductility  and  composition,  a  material 
of  the  quality  used  in  buildings  is  satisfactory  for  most  pur- 
poses. The  requirements  need  not  generally  be  as  severe  as 
for  bridge  steel,  although  the  wide  use  of  the  standard  bridge 
steel,  as  specified  by  the  Maintenance  of  Way  Association  and 
by  most  of  the  railroad  companies,  tends  to  facilitate  its  adop- 
tion for  all  structural  purposes. 

123.  Bond  Stress. — The  factor  of  safety  with  reference  to 
the  slipping  of  the  rods  should  be  at  least  3,  since  the  strength 
of  a  beam  should  not  be  limited  by  the  strength  of  bond.  From 
the  data  of  Chapter  IV,  we  may  take  the  bond  strength  of  plain 
steel  at  from  200  to  250  lbs/in2.  A  working  stress  of  from  60 
to  80  lbs/in2  is  therefore  suitable.  Increase  in  age  will  increase 
the  factor  of  safety  in  this  respect  very  considerably.  With  a 
working  bond  stress  of  60,  say,  and  a  tensile  unit  stress  of  15,000 
a  round  bar  will  need  to  be  embedded  a  length  of  15,000/4x60 
=  62.5  diameters  to  develop  its  full  strength.  In  the  case  of 
large  bars  of  1"  to  \\"  in  diameter  this  length  is  very  consid- 
erable and  for  short  beams  may  be  difficult  to  secure.  The 
deformed  bar,  or  the  anchored  bar,  is  of  especial  value  under 
these  conditions. 

For  deformed  bars  having  a  positive  grip,  a  working  stress 
of  150  lbs/in2  gives  an  ample  margin  of  safety.  A  larger  value 


§  123.]  BOND    STRESS.  217 

is  undesirable,  as  it  is  preferable  to  keep  the  stress  below  the 
ultimate  bond  strength  for  plain  bars  so  as  to  avoid  initial  slip 
under  working  conditions.  At  a  value  of  150  lbs/in2  the  re- 
quired length  of  embedment,  with  a  tensile  unit  stress  of  15,000 
lbs/in2,  would  be  25  diameters. 

In  calculating  bond  stress  the  method  of  Art  92  should  be 
used.  For  continuous  beams  the  shear  and  the  bond  stress  is 
a  maximum  near  the  support  and  it  also  changes  sign  at  the  sup- 
port. This  condition  gives  rise  to  a  sudden  change  in  the 
direction  of  bond  stress.  Thus  in  Fig.  576,  on  the  left  of  the 
support,  the  concrete  pulls  towards 
the  left  on  the  upper  rod,  and  on 
the  right  it  pulls  towards  the  right, 
as  shown  by  the  small  arrows. 
Any  slipping  increases  the  defor- 
mation of  the  concrete  at  once, 
and  hence  increases  the  tension  in 
the  concrete  at  the  center.  Like- 
wise, at  the  bottom,  any  slip  tends 
to  increase  the  compressive  stress 

in  the  concrete.     It  follows,  there- 

FIG.  576. 
fore,  that  where  rods  continue  over 

the  support  in  continuous  beams,  the  bond  stress  should  be 
fully  taken  care  of  on  each  side  of  the  center  of  support, 
otherwise  the  deformations  and  the  concrete  stresses  will  be 
increased.  For  compressive  reinforcement  the  formulas  of 
Art.  92a  may  be  used,  but  generally  it  will  be  sufficient  to 
consider  simply  the  maximum  compressive  stress  in  the  steel 
and  provide  a  sufficient  length  from  this  point  to  the  end  of 
the  bar  to  transmit  this  stress.  The  chance  for  an  end  bear- 
ing against  the  concrete  in  the  case  of  compressive  reinforce- 
ment reduces  the  danger  from  ultimate  failure  by  failure  of 
bond. 

Use  of  Anchored  Bars. — Large  bars  are  frequently  anchored 
at  their  ends  by  nuts  and  washers,  or  partially  anchored  by 
means  of  sharp  bends.  A  positive  anchorage  secures  a  reliable 


218  WORKING    STRESSES.  [Cn.  V. 

bond,  but  such  an  arrangement,  if  actually  brought  into  action, 
results  in  a  different  distribution  of  stress  than  where  adhesion 
is  depended  upon.  So  far  as  the  anchorage  is  effective  the 
resistance  in  the  concrete  is  furnished  by  so-called  "arch  action." 
In  the  ideal  case  of  anchorage,  where  the  bond  stress  is  zero 
and  the  full  load  comes  upon  the  anchorage,  there  will  be  com- 
plete arch  action.  There  will  be  no  horizontal  shear  and 
cracks  will  tend  to  form  nearly  vertically.  The  total  elonga- 
tion of  the  rods  will  be  greater  than  in  the  true  beam  and  the 
cracks  at  the  center  will  tend  to  open  up  wider,  throwing  the 
center  of  compression  higher  up.  The  compressive  stresses  in 
the  concrete  will  therefore  be  somewhat  higher  than  in  the  beam. 
In  practice  such  complete  arch  action  is  not  secured,  and  the 
effect  of  anchorage  upon  the  concrete  stresses,  either  tensile  or 
compressive,  need  hardly  be  considered.  Positive  anchorage 
may  well  be  employed  for  large  rods  where  the  length  is  insuffi- 
cient to  develop  the  necessary  bond  strength.  The  use  of  short 
square  hooks  upon  the  ends  of  bars  is  not  of  great  value.  The 
tests  of  Bach,  quoted  in  Art.  40,  show  that  the  square  bend  is 
not  very  effective  in  preventing  initial  slip  but  affords  a  con- 
siderable factor  of  safety  against  ultimate  failure.  If  hooks 
are  used  they  should  be  in  a  bend  of  180°,  as  explained  in  Art. 
40. 

124.  Shearing  Stresses. — From  the  results  discussed  in 
Chapter  IV  the  ultimate  shearing  strength  of  a  beam  having 
no  web  reinforcement  may  be  taken  at  about  100  lbs/in2, 
calculated  as  average  shearing  stress  on  the  cross-section. 
Inasmuch  as  a  failure  due  to  high  shearing  stresses  is  apt  to 
be  sudden,  the  factor  of  safety  should  be  at  least  three.  This 
gives  a  working  stress  of  about  30  lbs/in2.  For  beams  in  which 
the  web  is  well  reinforced  the  working  stresses  may  be  made 
3  or  4  times  as  great,  or  about  100  lbs/in2.  The  results  of 
tests  noted  in  Art.  110  show  that  in  spite  of  ample  web  rein- 
forcement visible  cracks  will  form  in  the  webs  of  beams  at 
maximum  shearing  stresses  about  equal  to  the  ultimate  tensile 
strength  of  the  concrete,  or  about  180  lbs/in2  in  the  beams 


§  125.]  WEB    REINFORCEMENT.  219 

tested.  The  working  stresses  should  therefore  keep  well 
within  this  limit,  and  as  the  maximum  shearing  stress  is  about 
15%  more  than  the  average,  the  value  of  100  lbs/in2  would 
appear  to  be  about  the  maximum  permissible. 

The  stresses  here  considered  relate  to  shearing  stresses 
involving  large  diagonal  tensile  stresses.  Where  such  tensile 
stresses  are  not  developed  to  any  extent,  as  in  " punching" 
shear,  a  higher  value  may  be  employed;  but  as  it  is  almost 
impossible  in  practice  to  avoid  altogether  such  tensile  stresses 
it  is  not  advisable  to  greatly  increase  the  working  stresses 
above  the  maximum  value  of  100  lbs/in2  above  suggested. 
A  value  of  150  lbs/in2  should  not  be  exceeded.  This  gives  a 
factor  of  safety  of  about  6  relative  to  the  shearing  strength,  as 
shown  in  Art.  22. 

125.  Calculation  of  Web  Reinforcement. — General  Condi- 
tions.— Before  considering  in  detail  the  calculation  of  web 
reinforcement,  or  reinforcement  against  inclined  tensile  stresses, 
the  reader  is  referred  again  to  the  discussion  in  Art.  46  and 
Fig.  12,  showing  the  lines  of  maximum  tension  in  a  homo- 
geneous beam.  In  the  reinforced  beam  the  intensity  of  the 
shearing  stress  is  nearly  uniform  from  the  neutral  axis  down 
to  the  horizontal  steel,  so  that  the  direction  of  maximum  ten- 
sion in  the  concrete  is  considerably  inclined  immediately  above 
the  steel.  This  inclination  is  greater  the  greater  the  shear 
and  the  less  the  horizontal  tension.  It  will  therefore  increase 
from  the  center  towards  the  end,  being  45°  where  the  hori- 
zontal tension  becomes  zero.  From  these  considerations  the 
ideal  web  reinforcement  would  be  a  system  of  rods  arranged 
somewhat  as  shown  in  Fig.  57c,  attached  at  their  lower  ends 
to  the  horizontal  rods,  or  consisting  of  numerous  horizontal 
rods  bent  up  as  indicated.  The  figure  also  indicates  roughly 
the  manner  in  which  the  inclination  of  diagonal  cracks  near 
the  bottom  tends  to  vary  from  nearly  vertical  at  the  center  to 
a  large  inclination  at  the  end.  The  exact  conditions  depend 
upon  the  nature  of  the  loading,  concentrated  loads  tending  to 
extend  the  region  of  large  shear  to  greater  distances  from  the 


220 


WORKING    STRESSES. 


[On.  V. 


support.  Generally  speaking,  then,  the  ideal  web  reinforce- 
ment should  have  greater  inclination  near  the  support  than 
near  the  center.  It  is  also  evident  that  to  be  effective  it  is 
necessary  that  it  be  spaced  at  relatively  close  intervals  and 
that  it  is  chiefly  effective  in  the  region  below  the  neutral  axis. 
Attention  is  also  called  to  the  discussion  of  Art.  90  and  the 
tests  of  Art.  109,  showing  the  effect  of  low  unit  stresses  in  the 
horizontal  steel  in  reducing  the  deformations  and  the  tendency 
to  the  formation  of  inclined  cracks.  This  condition  makes  it 
desirable  to  extend  a  considerable  part  of  the  horizontal  rein- 
forcement to  the  end  of  the  beam:  and  if  some  of  the  rods  are 

I 


FIG.  57c. 

bent  up,  the  bends  should  be  made  somewhat  beyond  the 
theoretical  points  required  for  bending  moment,  so  that  the 
actual  working  stresses  in  the  horizontal  steel  near  the  end  of 
the  beam  will  be  low. 

In  practice,  the  method  of  reinforcement  indicated  in  Fig. 
57c  cannot  well  be  used.  The  number  of  horizontal  rods  is 
generally  much  too  small  and  it  is  not  convenient  to  handle  the 
rods  when  bent  up  at  a  greater  number  of  points  and  at  various 
inclinations.  Instead  of  this  arrangement,  various  methods, 
as  illustrated  in  Art.  107,  are  employed,  the  common  practice 
being  to  use  a  few  bent  rods  combined  with  vertical  stirrups. 
From  the  considerations  of  the  preceding  paragraph  it  would 
appear  that  rods  bent  at  a  moderate  angle  would  be  well  suited 
for  sections  near  the  center  and  vertical  stirrups  for  sections 
near  the  end  where  the  reinforcing  members  must  be  spaced 
closer  together  and  at  greater  inclinations;  This  accords  gen- 
erally with  the  best  practice. 


§125]  WEB    REINFORCEMENT  221 

Length  of  Horizontal  Bars. — In  determining  the  length  of 
the  various  horizontal  bars  necessary  to  resist  the  bending 
moment,  the  same  method  may  be  used  as  in  the  design  of 
plate  girder  flanges.  If  the  bending  moment  is  due  to  uniform 
load  the  parabolic  formula  may  be  used,  as  explained  in  John- 
son's " Modern  Framed  Structures,"  Chapter  XIX.  It  is 


I 

v/  4. 

in  which  xn  =  length  of  the  nth  rod  in  the  order  of  length,  count- 
ing the  shortest  as  number  one; 
i  =  length  of  span ; 
A ••=  total  steel  area  at  center;  and 
di,  02,  etc.    =area  of  each  rod  up  to  the  nth  rod. 

For  unsymmetrical  loading  the  maximum  moments  at 
various  sections  will  need  to  be  determined  and  the  lengths 
obtained  therefrom. 

Web  Reinforcement. — Sufficient  experimental  work  has  not 
been  done  to  enable  the  proportioning  of  web  reinforcement  to 
be  done  with  any  degree  of  exactness.  However,  a  rough 
estimate  of  the  requirements  can  be  determined  on  rational 
grounds.  The  tests  already  quoted  in  Chapter  IV  indicate 
that  beams  with  horizontal  bars  only  cannot  be  stressed  safely 
beyond  about  30  lbs/in2  average  shearing  stress,  the  strength 
depending  on  the  quality  of  concrete  and  the  unit  stresses 
adopted  for  the  horizontal  steel.  In  practice  it  will  rarely 
happen  that  a  beam  need  carry  more  than  100  lbs/in2  average 
shearing  stress,  and  tests  of  the  best  work  indicate  that  this 
should  be  about  the  maximum  limit,  even  with  an  effective 
system  of  web  reinforcement. 

Where  it  becomes  necessary  to  provide  web  reinforcement, 
and  the  shearing  stresses  exceed  a  safe  limit  of  say  30  lbs/in2 
on  the  concrete,  some  estimate  must  be  made  of  the  stresses 
in  the  steel,  and  an  important  question  arises  as  to  the  mutual 
action  of  concrete  and  steel,  and  whether  the  concrete  can 


222 


WORKING    STRESSES. 


[Cn.  V. 


still  be  counted  upon  for  its  safe  stress  or  whether  the  steel 
must  be  proportioned  to  carry  the  entire  load.  In  this  con- 
nection certain  tests  made  by  Mr.  Withey  are  instructive.*  In 
the  case  of  two  rectangular  beams  and  two  T-beams  reinforced 
by  horizontal  bars  and  vertical  stirrups,  failure  was  caused  by 
the  overstressing  of  the  stirrups,  in  two  cases  the  stirrups  break- 
ing. The  results  were  as  follows: 


No.  of 
Beam. 

Cross-section 
of  Stirrup. 
Sq.  in 

Spacing  of 
Stirrups. 
Ins 

Net  Depth 
of  Beam. 
Ins. 

Width 
of  Beam. 
Ins. 

Av.  Vert. 
Shearing 
Stress  (t/). 
Lbs/in2. 

Calculated 
Stress  in 
Stirrup. 
Lbs/in2. 

GI 

.049 

5* 

13^ 

8 

222 

100,000 

G, 

.049 

6* 

13* 

8 

223 

100,000 

MI 

.049 

6 

16 

8 

272 

133,000 

M2 

.049 

6 

16 

8 

235 

115,000 

The  stirrups  were  single  loops  of  J-inch  round  steel  having  a 
yield-point  of  47,000  lbs/in2  and  an  ultimate  strength  of  62,000 
lbs/in2.  The  arrangement  of  rods  in  beams  M  is  shown  in 
Fig.  50a.  The  " calculated  stresses"  were  determined  on  the 
assumption  that  the  stirrups  carried  the  vertical  shear  in  a 
length  equal  to  the  depth  of  the  beam,  thus  causing  a  stress 
in  each  stirrup  equal  to  v'bs,  where  s  =  spacing  of  stirrups.  As 
these  stresses  are  much  beyond  the  ultimate  strength  of  the 
stirrups  it  is  evident  that  a  large  amount  of  shear  (about  40%) 
was  carried  by  the  concrete  and  by  the  bending  resistance  of 
the  horizontal  rods.  Tests  on  beams  without  stirrups  show  the 
average  shearing  strength  of  concrete  to  be  about  100  lbs/in2, 
indicating  that  approximately  correct  results  would  be  reached 
if  the  concrete  be  assumed  to  carry  its  full  value  and  the  stir- 
rups the  remainder.  Similar  results  have  been  reached  by 
other  experimenters.  If  this  is  true  at  ultimate  loads,  it  would 
be  even  more  certain  at  working  loads  where  the  concrete  is 
only  slightly  cracked  at  most  and  the  distribution  of  stress 


*  Bull.  No.  2,  Vol.  4,  1908,  Univ.  of  Wis. 


§  125.] 


WEB    REINFORCEMENT. 


223 


more  normal.  From  these  considerations  we  may  conclude 
that  in  calculating  stresses  in  web  reinforcement  the  concrete 
may  be  assumed  to  carry  its  safe  load  and  the  steel  proportioned 
to  carry  the  remainder.  These  stresses  may  be  estimated  in 
the  following  manner: 

In  Fig.  57d  are  represented  two  types  of  web  reinforcement, 
vertical  rods,  and  rods  inclined  at  45°.  Let  s  represent  the 
horizontal  spacing  in  both  cases  and  assume  the  line  of  failure 
at  45°.  Let  V  represent  the  shear  not  carried  by  the  concrete. 
Assume  for  simplicity,  that  the  intensity  of  shear  is  uniform 
over  the  section  and  is  equal  to  V/bd  =  v'.  This  will  also  be 
taken  as  the  intensity  of  the  diagonal  tensile  stress  at  45°. 
(The  maximum  will  be  only  12%  to  15%  more  than  this,  see 


IttJ 


FIG.  57d. 


Art.  89.)  In  the  case  of  vertical  stirrups  they  will  be  called 
upon  to  carry  the  vertical  component  only  of  this  diagonal 
tension,  the  horizontal  component  being  carried  by  the  hori- 
zontal bars.  This  vertical  component  per  unit  of  horizontal 
area  will  also  be  vf.  Assuming  equal  stresses  in  each  stirrup, 
represented  by  P,  we  have  finally 


V 

P=v'bs=-j--  s. 


(2) 


For  inclined  stirrups  the  most  unfavorable  assumption  is  that 
they  carry  the  full  diagonal  tension.  The  spacing  at  right 
angles  to  the  line  of  rupture  is  s  cos  45°,  and  the  stress  will 
therefore  be 


P  =V&s  cos  45°  =  0.7-7- 
a 


(3) 


224  WORKING    STRESSES.  [Cn.  V. 

The  foregoing  calculations  must  be  considered  as  only 
roughly  approximate,  but  they  are  doubtless  on  the  safe  side 
and  are  on  a  rational  basis.  In  preventing  initial  rupture  and 
distortions  under  low  loads  the  inclined  reinforcement  is  more 
effective  than  vertical,  as  it  receives  stress  at  an  earlier  stage, 
but  in  resisting  larger  distortions  the  vertical  type  appears  to 
be  equally  good.  Their  relative  efficiency,  however,  depends 
largely  upon  other  elements,  such  as  available  bond  strength, 
closeness  of  spacing  and  other  practical  considerations. 

Spacing  and  other  Details. — To  be  reasonably  effective  the 
web  reinforcement  should  be  so  spaced  that  at  least  one  rod  will 
intersect  any  45°  line  of  rupture  below  the  center  of  the  beam. 

As  shown  by  the  sketch  (Fig.  57 'e)  this  requires  a  spacing 
of  vertical  reinforcement  not  greater  than  d/2,  and  for  diag- 
onal rods,  a  horizontal  spacing  not 
greater  than  d.  Some  advantage  is 
gained  by  rods  spaced  somewhat 
further  apart,  but  tests  by  Talbot 
show  little  or  no  value  in  vertical 
rods  spaced  a  distance  apart  about 
equal  to  d.  The  working  stress 

to  be  used  in  these  calculations  should  be  no  higher  than  per- 
mitted elsewhere,  and  preferably  lower,  as  it  is  desired  to  pre- 
vent large  distortions  so  far  as  practicable.  In  bending  up 
horizontal  rods  those  remaining  straight  should  be  ample  in 
number  to  take  the  moment  stresses,  and,  preferably,  at  reduced 
intensities  towards  the  ends  of  the  beams.  A  small  angle  of 
bend  near  the  center  and  larger  angles  near  the  end,  as  in 
Fig.  57c,  should  be  observed  so  far  as  practicable.  It  will 
often  be  impracticable  to  provide  as  much  reinforcement  as 
desired  by  means  of  bent-up  rods,  and  some  vertical  stirrups 
will  be  needed,  especially  near  the  end  where  the  stresses  are4 
high.  A  combination  of  bent  rods  and  vertical  stirrups  is 
common  practice  and  readily  lends  itself  to  adequate  and  con- 
venient treatment.  For  large  beams,  under  heavy  shearing 
stresses,  both  should  be  used. 


§  126.]  WEB   REINFORCEMENT.  225 

Inclined  stirrups  are  quite  as  effective  as  bent  rods  or  ver- 
tical stirrups,  but  to  prevent  slipping  on  the  horizontal  rods  they 
should  be  securely  fastened  thereto.  For  example  see  Art.  135. 

In  detail,  stirrups  may  be  made  in  various  forms,  as  indi- 
cated in  Fig.  50.  Woven  wire,  bent  around  the  rods,  is  a 
satisfactory  and  very  effective  reinforcement.  Where  com- 
pressive  bars  are  used  in  the  upper  part  of  a  beam  the  stirrups 
should  hook  around  these  bars  also.  In  continuous  beams  the 
upper  face  becomes  the  tension  face  near  the  supports  and 
this  is  also  where  the  shear  is  large.  Stirrups  in  this  vicinity 
should  loop  about  the  upper  bars. 

The  bond  strength  of  web  reinforcement  must  be  carefully 
guarded,  especially  in  the  case  of  large  bent-up  rods.  This 
strength  should  be  provided  in  the  upper  portion  of  the  beam. 
Plain  rods  bent  up  often  lack  sufficient  bond  strength  to  render 
them  fully  effective.  Where  bent  up  at  a  considerable  angle 
they  should  be  turned  again  horizontally  and  extend  some 
distance  along  the  upper  part  of  the  beam  as  in  Fig.  (6),  p.  159. 
In  non-continuous  beams  the  ends  of  the  bars  should  be  bent 
into  a  hook. 

The  following  simple  graphical  method  may  be  used  in 
important  cases  (large  beams)  for  determining  the  stress  or 
spacing  of  bent  rods.  It  also  serves  to  make  clear  the  princi- 
ples involved.  Suppose  OB,  Fig.  57/,  represent  half  of  a  simple 
beam  uniformly  loaded.  Calculate  the  shearing  stress  vf  at  the 
end.  Project  the  axis  OB  upon  an  axis  OC  at  45°  inclination 
and  lay  off  CC'  equal  to  v't  and  draw  the  line  OC'.  Then  the 
ordinates  between  OC  and  OC'  will  represent  the  shearing 
stresses  v'  along  the  beam;  and  the  area  between  any  two 
ordinates  DD'  and  EE' ,  multiplied  by  the  width  b  of  the  beam, 
will  equal  the  product  of  the  total  average  shear  over  the  length 
s,  times  the  projection  of  this  length  on  the  inclined  axis  OC. 
It  will  represent  therefore  the  stress  in  a  rod  bent  up  at  45°  at 
point  G,  in  line  with  the  center  of  gravity  of  the  area  DEE'D'. 
For,  by  eq.  (3),  this  stress  is  equal  to  0.7Vs/d  =  0.7  v'bs.  But 
Q.7v  's  is  the  area  DEE'D' ',  v'  being  the  value  of  the  average 


226 


WORKING    STRESSES. 


[Cn.  V. 


ordinate.     Hence    Ps  =  area  DEE'D'Xb.      If    rods    are    bent 
FIG.  57 f.  at  other  angles,  then  the  axis  OC 

J  may  be  drawn  at  right  angles 

|'|  thereto.     If  the  concrete  be  as- 

sumed to  carry  a  portion  of  the 
B  shear  vf  and   stirrups   another 
A.      constant  portion,  these  amounts 
I      may  be  subtracted  graphically 
from  the    total  shear  as  indi- 
cated in  Fig.  57g,  and  the  re- 
mainder taken  by  bent  rods.    If 
s  =  spacing  of  stirrups  and  P= 
safe    stress,    then    the   amount 
taken  by  stirrups  will  be  found 
from  eq.  (2)  equal  to  v'  =  Pss/b. 
In  no  case  should  the  area  of  the 
horizontal  rods  be  taken  into  ac- 
count .  These  offer  resistance  only 
FIG.  570.  through  their  bending  strength. 

126.  Spacing  of  Bars. — In  rectangular  or  T-beams  the  spac- 
ing of  bars  is  important;  in  T-beams  this  consideration  will 
largely  control  the  width  of  the  beam.  The  requirement  in 
general  as  to  spacing  is  that  the  amount  of  concrete  left 
between  the  bars  must  be  sufficient  to  transmit  to  the  upper 
part  of  the  beam  the  stress  which  the  bars  give  over  to  the 
concrete  below  them.  If  the  bars  are  circular  it  may  be  assumed 
that  one-half  of  the  stress  in  them  is  given  over  to  the  concrete 
below,  hence  the  strength  of  the  concrete  on  a  longitudinal 
section  through  the  center  plane  of  the  bars  must  equal  one- 
half  of  the  stress  in  the  bars.  If  the  shearing  stress  be  taken 
as  equal  only  to  the  bond  stress  then  the  clear  space  between 
bars  must  be  one-half  the  circumference  of  a  bar,  or  1.57 
diameters.  In  the  sense  here  employed  the  shearing  strength 
is  at  least  twice  the  bond  strength  for  smooth  rods,  so  a  clear 
spacing  of  less  than  one  diameter  is  sufficient  from  this  stand- 
point. In  the  case  of  square  bars,  on  the  same  basis,  the  clear 
spacing  would  need  to  be  1J  diameters  if  the  bars  are  placed 
with  sides  vertical,  or  one  diameter  if  placed  with  sides  diagonal. 


§  127.]  ECONOMICAL   WORKING    STRESSES.  227 

But  in  addition  to  the  shearing  stresses  there  is  likely  to  be 
developed  more  or  less  tension  in  the  concrete  surrounding  the 
rods,  so  thatt  here  should  be  left  ample  areas  of  concrete  between 
them,  especially  towards  the  end  where  the  bond  stresses  are 
large.  The  space  should  also  be  sufficient  to  permit  satisfac- 
tory manipulation  of  the  concrete.  A  minimum  clear  spacing 
of  at  least  1J  diameters  should  be  provided,  with  an  equal 
distance  between  the  outside  rod  and  the  surface  of  the  beam. 
Where  some  of  the  rods  are  bent  up  the  spacing  can  readily 
be  made  more  liberal  towards  the  end  of  the  beam.  Between 
two  horizontal  layers  of  rods  the  spacing  may  be  less  but  should 
be  sufficient  to  insure  good  bond. 

Liberal  spacing,  or  large  net  section  of  concrete,  favors  large 
rods  and  few  in  number;  good  bond  strength  without  waste  of 
material  favors  small  rods.  If  bent  rods  are  to  be  used  for  web 
reinforcement,  then  numerous  small  rods  are  also  advantageous. 
If  the  bond  strength  is  not  in  question,  or  can  easily  be  taken 
care  of,  then  large  rods  are  desirable,  but  more  stirrups  or  other 
secondary  reinforcement  may  be  needed  than  where  small  rods 
are  used. 

127.  Economical  Proportions  and  Working  Stresses. — For 
given  unit  prices,  the  cost  of  concrete  beams  per  unit  of  resisting 
moment  will  vary  with  the  proportions  adopted  for  breadth  and 
depth,  and  with  the  working  stresses  employed.  -Because  of 
the  mutual  relations  between  the  concrete  and  steel  it  may 
happen  that  the  maximum  economy  of  construction  may  be 
obtained  by  using  less  than  the  allowable  working  stresses  in 
one  or  the  other  of  the  two  materials.  It  will  therefore  be 
useful  to  investigate  the  effect  on  cost  of  variations  in  propor- 
tions and  in  the  working  stresses. 

Consider  a  portion  of  a  rectangular  beam  one  unit  in  length. 

Let  c  =  cost  of  concrete  per  unit  volume;  r  =  ratio  of  cost  of 
steel  to  cost  of  concrete  per  unit  volume ;  p  =  ratio  of  steel  area 
to  concrete  area;  C  =  cost  of  beam  per  unit  length. 
Then  C  =  c(bd  +  rpbd)  =  cbd(1.-}-rp).        ....    (1) 

From  Art.  59  we   have  M2=M/R,  in  which  M  =  bending 


228 


WORKING  STRESSES. 


[CH.  V. 


moment  and  R  =  coefficient  of  strength  of  the  beam,  depending 
in  value  only  upon  /5,  fc,  and  n.  From  this  we  may  write 
bd=M/*Rd,  bd=VMb/R,  and  bd=V(b/d)(M2/R2),  whence  we 
derive  the  three  expressions  for  cost  : 

M 

>...'....    (2) 


Mb 
R' 


and 


(3) 


(4) 


128.  General  Effect  of  Varying  Proportions. — Since  the  values 
of  R  and  p  depend  only  on  f8,  fc  and  n  we  note  from  (2)  that  the 
cost  of  a  rectangular  beam  to  support  a  given  moment,  Mt  varies 
inversely  with  the  depth;  and  from  (3)  that  the  cost  varies 
directly  with  \/breadth;  and  finally,  from  (4)  that  it  varies 
with  the  cube  root  of  the  ratio  of  breadth  to  depth.  In  all  cases 
it  is  assumed  that  the  two  dimensions  are  made  to  correspond 
with  each  other  as  calculated  from  the  selected  values  of  /a 
and  fc.  It  follows  from  (2)  that  with  given  values  of  /.  and  /«. 
the  deeper  the  beam  the  less  the  cost,  so  long  as  b  can  be  reduced 
accordingly.  The  depth  will,  however,  be  limited  in  various 
ways.  It  may  be  limited  by  the  requirement  of  shearing  stress 
fixing  the  value  of  bd,  or  it  may  be  limited  by  the  head  room 
required,  or  it  may  practically  be  limited  by  the  fact  that  a 
certain  breadth  is  necessary  to  give  a  convenient  and  proper 
covering  of  the  steel  reinforcement  or  to  give  a  beam  of  satis- 
factory proportions.  In  the  construction  of  continuous  surfaces, 
such  as  floor  slabs,  the  case  is  one  of  fixed  width,  since  the  width 
of  beam  to  carry  the  load  coming  upon  a  strip  one  foot  wide 
is  also  one  foot.  We  may  then  consider  four  cases  according 
to  the  particular  feature  of  the  design  which  is  the  controlling 
element.  These  cases  are: 

(a)  When  the  area  of  cross-section  is  determined  by  the 

shear ; 

(b)  When  the  depth  of  the  beam  is  fixed : 


§  130.]  ECONOMICAL  WORKING  STRESSES.  229 

(c)  When  the  width  of  the  beam  is  fixed; 

(d)  When  the  ratio  of  width  to  depth  is  fixed. 

129.  (a)  The  Area  of  Cross-section  is  Determined  by  the 
Shear.  —  A  given  value  for  shearing  stress  requires  a  fixed  value  of 
bd,  but  the  requirement  for  bending  moment  is  that  bd  =  M/Rd', 
hence  if  a  beam  is  designed  for  moment  alone  the  area  bd  will  be 
less  the  deeper  the  beam.  Theoretically,  therefore,  for  a  given 
value  of  R  the  maximum  depth  permissible  is  that  for  which  the 
resulting  area  bd  is  just  large  enough  to  carry  the  shear.  If  V  is 
the  total  shear  and  v'  is  the  permissible  shearing  stress,  then  bd  = 
V/tf.  Also  bd  =  M/Rd.  Hence  for  equal  strength  M/Rd  =  V/v' 
and  therefore 

......    .    .    .    (5) 


and  b  =  V/v'd.    ........     (6) 

These  equations  give  the  dimensions  of  a  beam  which  will 
be  of  just  the  required  strength  in  moment  and  shear.  It  re- 
mains to  be  determined,  however,  whether  a  still  greater  depth 
will  result  in  greater  economy. 

If  a  greater  depth  be  used,  bd  must  remain  constant;  hence 
bd2  will  be  increased  and  the  concrete  stress,  /c,  decreased. 
Reference  to  Plate  III,  p.  2T3,  shows  that  with  constant  /«,  a  de- 
crease in  the  value  of  fc  permits  the  use  of  a  smaller  percentage 
of  steel.  Hence  with  increasing  depth  and  constant  bd  (or 
volume  of  concrete),  the  amount  of  steel  will  be  reduced,  and 
therefore  the  cost.  The  proportions  of  the  beam  will  therefore 
not  be  determined  by  the  shear  excepting  as  to  minimum  cross- 
section. 

130.  (6)  The  Depth  of  the  Beam  is  Fixed.  —  From  eq.  (2)  it 
is  seen  that  for  given  values  of  M  and  d  the  cost  varies  with 
(l  +  rp)/R.  Now  p  and  R  depend  only  upon  the  working  stresses 
/,  and  }c  (n  being  constant),  hence  it  will  be  convenient  to 
determine  the  variation  in  cost  due  to  variation  in  f8  and  /c, 
assuming  certain  values  for  r.  Results  of  this  analysis  are 
shown  in  Fig.  58  for  values  of  r  of  60  and  80  and  for  various 
values  of  f8  and  fc.  The  results  are  very  instructive  and  show 


230 


10000 
.028 


.026 


.021 


.018 


.016 


.011 


g   .010 
53    .030 

•^      AOQ 


.026 
.024 
.022 
.020 
.018 
.016 
.014 
.012 


WORKING  STRESSES. 

12000  14000  16000  18000 


=  40 


SO 


10000  12000  14000  16000  18000 

FIG.  58.— Relative  Cost  for  Fixed  Depth. 


[Cn.  V. 

20000 


.024 


.022 


.020 
.018 
.016 
.014 
.012 
.010 


.024 
.022 
.020 
.018 
.016 
.014 
.012 


20000 


§  133.]  ECONOMICAL  WORKING  STRESSES.  231 

that  for  values  of  fc  of  500  or  600  lbs/in2  no  economy  is  secured 
by  using  values  of  fs  greater  than  12-14,000  lbs/in2.  For  larger 
values  of  r  or  of  }c,  higher  values  can  economically  be  used  for  Jst 
but  a  value  of  80  for  r  is  not  likely  to  be  greatly  exceeded.  If  the 
cost  of  concrete  be  as  low  as  20  c.  per  cu.  ft.  the  corresponding 
cost  of  steel  would  be  $16.00  per  cu.  ft.,  or  3.2  c.  per  pound. 
This  is  a  low  cost  of  concrete  and  a  high  cost  of  steel.  The  dia- 
gram shows  that  the  cost  is  decreased  by  increased  values  of  fc. 

131.  (c)  The  Width  of  the  Beam  is  Fixed.— From  eq.  (3) 
the  cost  for  given  values  of  M  and  b  varies  with  (l+r/?)/V#. 
Fig.  59  represents  this  quantity  plotted  for  various  values  of 
f8  and  fc.    Comparing  this  with  Fig.  58  it  is  seen  that  somewhat 
higher  values  of  fs  are  warranted,  but  it  is  evident  that  the 
gain  in  economy  is  very  small  for  values  above  16,000  lbs/in2, 
except  where  the  steel  is  very  expensive  and  the  concrete  cheap. 

132.  (d)  The  Ratio  of  Width  to  Depth  is  Fixed.— It  is  often 
desired  to  secure  approximately  a  certain  given  ratio  of  breadth 
to  depth.    In  this  case  we  find  from  eq.  (4)  that  the  cost  will 
vary  with  (l  +  rp)/RL    Fig.   60   represents  this   quantity  for 
various  values  of  fs  and  fc.    It  is  seen  that  the  most  economical 
values  will  lie  between  those  of  cases  (6)  and  (c). 

133.  Floor  Slabs  with  Weight  of  Concrete  Eliminated. — In 
all  the  foregoing  discussion  the  moment  to  be    resisted    has 
included  that  due  to  the  weight  of  the  beam  itself.    For  large 
beams  and  girders  this  is  unimportant  in  this  connection,  but 
with  floor  slabs,  where  the  external  load  is  small,  the  weight  of 
the  material  itself  modifies  the  results  to  a  large  extent.     General 
results  cannot  be  presented  for  all  cases,  but  the  analysis  will  be 
given  for  a  single  case  represencing  ordinary  conditions.    A 
span  length  of    10  ft.  has  been  taken  and  a  net  floor  load  of 
150  lbs/ft2.    Then  from  Table  No.  21,  Chap.  VI,  the  required 
cross-section  and  amount  of  steel  has  been  determined  for  vari- 
ous values  of  fs  and  fc.    The  relative  cost  per  unit  floor  area  has 
been  calculated  for  values  of  r  of  40,  60,  80,  and  100  and  the  re- 
sults plotted  in  Fig.  61.     Comparing  these  results  with  those  of 
Fig.  59,  where  the  weight  of  the  beam  has  not  been  deducted, 


52 

10 
.1800 

,1750 
,1700 
.1650 
.1600 
.1550 
.1500 
.1450 

.1400 

.2000 

J.1950 
|.1900 
W.1850 
.1800 
.1750 
.1700 
.1650 
.1600 
.1550 
.1500 
1450 

WORK 

000                 12000                  14 

ING  STRESSES. 

WO                 16000                  18000                 200 

[Cn.  V. 

00 
.1800 

.1750 
.1700 
.1650 
.1600 
.1550 
.1500 
.1-450 
.1-400 

.1350 

.1300 
.1900 

.1850 
.1800 
.1750 
.WOOL 
.1650 

.1600 
.1550 

.1500 
.1450 

.1'400 
)0 

X 

fa 

^ 

^^ 

f 

=  400 

—  • 

- 

^gf^^ 

V 

\ 

v\ 

X 

r*= 

=  60 

\ 

\ 

^ 

^ 

y» 

X^ 

x 

m  ^^ 

**  

—  —  . 

- 

—  —  —  —  . 

\s 

^< 

5^ 

«___— 

V 

•MBWMM 

MM^M^M 

-_»_— 

X 

"^^ 

^-^ 

^-  

—  ^.^ 

s 

\ 

^ 

•\ 

N 

\ 

\ 

\ 

X 

^ 

^^ 

-- 

- 

—  

~~~** 

^ 

V^ 

N 

X 

x,^> 

r. 

^80 

V 

^ 

^ 

\ 

s 

"\&0 

^•^. 

^, 

-  —  — 

N, 

S^9 

X 

\ 

X^ 

2^, 

"S 

\ 

.1400 

A 

§ 

10000                  12000                 1"4000                  16000                  18000                  200 

FIG.  59.-— Relative  Cost  for  Fixed  Width. 


133.] 

.icir 


.095 


,090 


.080 


,075 


.070 


.OC5 


.060 


.055 
.105 


'.100 


.000 


.075 
,070 
.065 
,060 


ECONOMICAL  WORKING  STRESSES. 

12000         14000         1GOOO        18000 


233 


f. 


=400 


GO 


20000 


10000        12000        UOOO        16000        18000        20000 

FIG.  60.— Relative  Cost  for  Fixed  Ratio,  Breadth  to  Depth 


WORKING  STRESSES. 


fc 


400 


700 


r==  40 


/c=500 


,=  =600 


^400 


r=  60 


fc  =  i 


600 


f. 

^12000  13000  14000  15000  16000  17000  18000 

FIG.  61a.— Relative  Cost  for  Fixed  Breadth,  Weight  of  Beam  Deducted. 


§  135.] 


ECONOMICAL  WORKING  STRESSES. 


235 


12000 


13000 


1.05 
1.00 

.95 
.90 
.85 
.80 
.75 

LTO 

| 
I 

[ 

Yio 

1.05 
1.00 

.95 
.90 
.85 
.80 
J75 
.70 


14000 


15000 


J6000 


17000 


18000 


— 100 


r=80 


1.05 
1.00 

.95 
.90 

.85 
.80 
.75 


1.10 


l.OS 


t.oo 


12000      13000      14000       15000      16000       17000      18000 

FIG.  616.— Relative  Cost  for  Fixed  Breadth,  Weight  of  Beam  Deducted. 


236 


WORKING  STRESSES. 


[Cn.  V. 


it  is  seen  that  the  economical  values  of  /«  are  considerably  less. 
For  values  of  r  not  exceeding  60  and  for  fc  not  exceeding  £00 
there  is  no  reason  for  using  a  value  for  f8  higher  than  14,000 
lbs/in2.  For  other  spans  and  floor  loads  the  results  will  be 
somewhat  different,  but  the  variation  wijjjiot  be  great.  Larger 
floor  loads  and  shorter  spans  will  give  results  more  nearly  ap- 
proaching those  of  Fig.  59;  smaller  loads  and  longer  spans  will 
tend  in  the  opposite  direction. 

Percentages  of  steel  corresponding  to  any  particular  values 
of  fc  and  f8  are  given  by  reference  to  Plate  III,  p.  215. 

134.  Effect  of  Overlapping  Bars. — In  most  cases  the  rein- 
forcing bare  of  slabs  are  made  to  overlap  more  or  less;   where 
negative  moment  over  the  beams  is  taken  care  of  this  over- 
lapping may  be  25  to  30  per  cent.    To  take  account  of  this 
in  using  the  equations  or  diagrams  of  the  preceding  articles, 
the  most  convenient  method  is  to  increase  the  unit  cost  of  steel, 
or  the  ratio  r,  by  the  same  percentage  that  measures  the  over- 
lap of  the  steel. 

135.  T-Beams. — General  Design. — T-beams  occur  in  prac- 
tice generally  where  a  floor  slab  and  beam  are  built  as  a  mono- 
lithic structure,  as  in  floor  construction.     Occasionally,  also, 
where  heavy  girders  are  required  it  is  expedient  to  design  the 
beam  in  the  form  of  a  T.     Inasmuch  as  the  only  purpose  of  the 
concrete  below  the  neutral  axis  is  to  bind  together  the  tension 
and  compression  flanges,  its  section  is  determined  by  the  shear- 
ing stresses  involved,  and  a  considerable  saving  can  thus  often 
be  effected  over  the  rectangular  form.     Where  the  flange  is  a 
part  of  the  floor  slab  its  thickness  is  already  determined  by 
other  considerations,  but  the  width  of  slab  which  can  be  taken 
as  effective  flange  width  must  be  estimated.     A  common  rule 
of  practice  is  to  count  a  width  of  slab  not  greater  than  one  third 
of  the   span  length,  but  this  should  in  fact  depend  also  upon 
thickness  of  slab  and  of  the  stem  of  the  T. 

If  made  too  wide  and  thin  the  shearing  stresses  along  the 
line  aa'  and  cc',  Fig.  62,  will  be  excessive  and  greater  than  those 
along  the  line  a'c'.  On  this  account  it  is  desirable  to  limit 


§  135.]  T-BEAMS.  237 

the  value  of  x  to  about  four  times  the  thickness  of  the  slab,  or 
three  times  the  width  of  beam  &'.  Experiments  show  that  a 
total  flange  width  of  3  or  4  times  the  width  of  the  web  gen- 
erally gives  ample  flange  area  so  that  the  design 
of  such  a  T-beam  consists  mainly  in  the  design 
of  the  web  or  stem,  and  the  proper  arrangement 
of  the  steel.  Of  course  the  width  of  flange 
cannot  exceed  the  spacing,  center  to  center,  of 
beams,  and  it  is  common  to  limit  the  effective 
width  to  three-fourths  of  the  spacing  of  beams. 

Where  a  T-beam  is  not  connected  with  a  floor  system  then 
the  size  of  flange  may  be  selected  to  meet  the  conditions  at 
hand.  In  this  case  the  stem  of  the  beam  should  first  be  deter- 
mined approximately,  on  1jie  basis  of  the  shearing  stresses  to 
be  carried.  A  suitable  flange  can  then  be  selected  by  a  few 
trials,  as  explained  in  Art.  73.  The  deeper  the  beam  the  less 
the  amount  of  steel  required  for  constant  cross-section.  But 
T-beams  should  not  be  made  too  deep  in  proportion  to  width, 
as  such  forms  are  relatively  weak  at  the  junction  of  stem  and 
flange.  All  re-entrant  angles  in  rigid  material  such  as  con- 
crete are  points  of  weakness  and  such  angles  should  therefore  be 
modified  by  curved  lines  or  be  made  obtuse  by  sloping  the  sides 
of  the  beam.  A  width  of  beam  sufficient  to  carry  the  shear  and 
to  give  plenty  of  space  for  the  bars  is  usually  ample.  The 
maximum  desirable  ratio  of  depth  to  width  may  be  taken  at 
about  two  for  small  beams  up  to  three  or  four  for  very  large 
and  massive  work.  Depths  are  often  determined  by  available 
head  room.  Beams  of  excessive  depths  are  objectionable  as 
being  more  difficult  and  troublesome  to  reinforce  properly; 
the  cost  of  web  reinforcement  also  becomes  relatively  greater. 
The  flanges  should  be  thorouglhy  bonded  to  the  web  by  means 
of  web  reinforcement  running  well  up  into  the  flange  arid, 
where  the  flange  is  wide,  by  additional  cross-reinforcement  in 
the  plane  of  the  flange.  (For  further  details  see  Art.  165). 

Economical  proportions. — Where  a  floor-slab  forms  the 
flange  of  a  T-beam,  then  the  economical  proportions  of  the 


238  WORKING    STRESSES.  [Cn.  V. 

stem  may  be  considered.  Here  the  slab  forms  practically  all 
the  compressive  area,  but  does  not  enter  into  the  cost  of  the 
beam.  Using  the  approximate  formula,  eq.  (7),  of  Art.  74 
the  area  of  the  steel  is  equal  to  M/fs(d'  +  $t),  in  which  df  is 
the  depth  of  beam  below  the  slab.  The  cost  is  then 


=c[W 


rM      "I 


From  this  expression  it  is  evident  that  the  cost  will  decrease 
with  increased  values  of  fa  under  all  conditions,  and  that  with 
a  fixed  value  of  b'd'  the  cost  decreases  with  increase  in  depth. 
If  d'  is  fixed  then  the  cost  will  be  a  minimum  when  &'  is  made 
as  small  as  possible,  and  its  value  will  then  be  determined  by 
the  shearing  stress  or  by  the  space  required  for  the  bars.  If 
the  value  of  b'  is  assumed  as  fixed,  then  there  is  a  definite 
value  of  d'  which  will  give  minimum  cost.  Considering  d'  as 
variable  and  V  as  constant  we  find  by  differentiation  that  for 
minimum  cost  the  value  of  d'  is  given  by  the  equation 


(2) 


From  this  expression  the  best  depth  for  various  assumed  widths 
can  readily  be  determined  and  the  desirable  proportions  finally 
selected. 

EXAMPLE  OF  THE  DESIGN  OF  A  T-BEAM.  —  To  illustrate  the  prin- 
ciples discussed  in  the  preceding  article  a  design  will  be  made  of  a 
large  girder  built  in  the  form  of  a  T-beam.  Assume  the  following 
data: 

Span  length  =40  ft.  ;  dead  load  =  1500  Ibs/ft.  ;  live  load  =  2500  Ibs/ft.  ; 
fc  =  600  Ibs/in.  ;  fa  =  15,000  Ibs/in.  ;  v'  =  30  lbs/in2  for  concrete  alone,  and 
100  lbs/in2  where  the  web  is  reinforced  against  tensile  stresses.  Bond 
stress  =  75  lbs/in2,  with  an  allowable  increase  of  50%  for  straight  rods 
near  the  end,  in  accordance  with  Art.  39.  The  beam  is  to  be  simply 
supported  at  the  ends  and  the  flange  is  to  be  proportioned  as  well  as 
the  web  ;  that  is,  the  flange  does  not  form  a  part  of  a  floo"  system  already 
determined. 


§  135.]  T-BEAMS.  239 

40,OOfx402Xl2 

Solution. — The     total   bending    moment,    M,    =  — 

o 

9,600,000  in-lbs.  The  maximum  shear,  V,  =4000X20  =  80,000  Ibs. 
The  required  net  web  area  =  b'd  =  80,000/100  =  800  in2.  This  can  be 
supplied  by  a  web  16"X50"  or  18"X45".  To  give  better  space  for 
the  steel  the  latter  will  be  chosen  for  a  preliminary  value.  A  thickness 
of  flange  of  12  in.  will  be  tried.  For  this  thickness  t/d=  12/45  =  .267. 
Then,  by  means  of  Plate  IX,  p.  283,  we  find  that  M/W  =  93,  bd2  = 
9,600,000/93  =  103,000,  and  6  =  103,000/452  =  51  in.  From  the  diagram 
the  value  of  /<*=. 89X45  =  40  in.,  and  ,4=9,600,000/40X15,000=16  in2. 
To  illustrate  the  effect  of  varying  proportions,  calculations  will 
also  be  made  for  a  .flange  thickness  of  8",  10",  14",  and  16".  The 
results  are  as  follows: 


t 

b 

id 

A 

Overhanging 
Width 
of  Flange. 

Area  of  Flange 
Outside  of 
Web. 

8  in. 

64  in. 

41  .  5  in. 

15.4  sq.  in. 

23  in. 

368  sq.  in. 

10  " 

56  " 

40.5  " 

15  ."8      " 

19  " 

380      " 

12  " 

51    " 

40.0  " 

16.0      " 

16*" 

396      " 

14  " 

50  " 

39.5  " 

16.2      " 

16  " 

448      " 

16  "          49  '  39.5  '  16.2  15*"  496      " 

It  will  be  noted  that  the  effect  of  variation  of  t  upon  the  amount 
of  steel  is  very  small,  but  that  the  amount  of  concrete  is  less  the 
thinner  the  slab.  The  saving  in  concrete  is  measured  by  the  reduc- 
tion in  the  areas  of  the  flange  exclusive  of  the  width  of  web.  The 
10-in.  flange  gives  16  sq.  in.  less  material  than  the  12-in.  and  the  8-in. 
12  sq.  in.  less  than  the  10-in.  At  the  same  time  considering  the  fact 
that  the  girder  is  not  a  part  of  a  floor  system  and  therefore  that  the 
flanges  are  unsupported  at  their  outer  edges,  and  also  that  some  trans- 
verse steel  will  be  required  to  bond  the  flange  well  together,  it  is 
evident  that  a  compact  beam  having  a  relatively  thick  flange  is 
desirable.  The  choice  would  probably  lie  between  the  10-in.  and  the 
12-in.  flanges.  The  12-in.  flange  will  be  adopted. 

The  steel  area  required  is  16  sq.  in.  This  will  be  made  up  of  five 
rods  If  in.  diameter,  and  seven  rods  lj  in.  diameter,  giving  a  total 
area  of  16.0  sq.  in.  To  provide  a  spacing  of  2^  diameters  the  rods 
will  be  placed  in  three  rows;  the  five  If  in.  rods  in  the  lower  row,  five 
1|  in.  rods  above,  and  two  l£  in.  rods  in  a  third  row.  In  bending  up 
the  rods  the  two  uppermost  rods  will  be  bent  up  nearest  the  center. 
The  arrangement  of  rods  in  cross-section  is  shown  in  Fig.  62a.  Taking 
moments  of  areas  about  the  center  of  the  lowest  row,  the  center  of 


240  WORKING    STRESSES.  [Cn.  V. 

5X1.23X2  +  21.23X4 
gravity  of  the  group  is  found  to  be  —  — — —          —  =1.4  in.  above 

this  row.  Hence  the  lower  row  should  be  placed  about  46£  in.  below 
the  top  of  the  beam,  thus  giving  a  total  depth,  in- 
cluding the  protective  covering  of  49  in. 

To   provide   shearing   reinforcement,    as    many    of 
the    rods   will   be    bent  up   as  practicable  considering 
their  necessary    lengths  for  resisting  the  bending  mo- 
ment.    This  length    may  be    found    by  a  diagram  of 
FIG   62a          maximum  moments,  or,  if  the  load  is  uniform  so  that 
the  moment  diagram  is  a  parabola,  by  use  of  eq.  (1), 
Art.   125.     Applying  this    equation   in  the  present  example,    we   have 
=  10.     The  necessary  lengths  of  the  rods  are  then  as  follows: 


No.  of  Rod  (ai  +  .  .  .  an)  z=10\(ai+  .  .  .  an) 

1  1.23  ll.lft. 

2  2.46  15.7  " 

3  3.69  19.5  " 

4  4.92  22.2  " 

5  6.15  24.8  " 

6  7.38  27.2  " 

7  8.61  29.3  " 

8  10.09  31.8  " 

The  rods  may  be  bent  up  at  any  point  beyond  the  required  lengths, 
as  given  above. 

k  Shearing  Stress  and  Reinforcement. — The  maximum  end  shear  is 
4000X20  =  80,000  Ibs.;  the  maximum  center  shear  is  2500X20Xi  = 
12,500  Ibs.  The  average  shearing  stresses  in  the  web  at  these  two 
points  are,  respectively,  100  lbs/in2  and  16  lbs/in2.  Under  the  speci- 
fications the  concrete  is  good  for  30  lbs/in2  without  reinforcement. 
Assuming  the  shear  to  vary  uniformly  from  center  towards  end  the 

value  of  30  lbs/in2  will  be  reached  at  a  distance  of  —  X  20  =  3.3  ft.  from 

84 

the  center.  Beyond  this  point  reinforcement  will  be  required. 
It  will  be  designed  on  the  assumption  that  the  concrete  may  be 
considered  as  carrying  30  Ibs  /in2  and  that  the  remainder  must 
be  carried  by  bent  rods  and  stirrups.  It  will  be  desirable  to  use 
relatively  low  working  stresses  in  the  steel  in  order  to  avoid  all 
danger  of  cracks. 

It  will  be  desirable  to  decide  first  upon  a  convenient  arrangement 
of  bent-up  rods.     Fig.  626  illustrates  such  an  arrangement,  which  pro- 


§  135.] 


T-BEAMS. 


241 


vides  lengths  somewhat  in  excess  of  those  required  for  bending  moment. 
It  spaces  bends  at  distances  apart  not  greater  than  the  depth  of  the 
beam  and  closer  near  the  end.  Two  rods  are  bent  up  at  each  place 
(to  bend  the  rods  singly  complicates  the  handling  materially).  For 
the  same  reason  the  bends  are  all  made  at  45°  and  the  bent  ends  are 
extended  far  enough  to  give  ample  strength  of  bond.  The  resulting 


FIG.  626. 


lengths  of  straight  portions  are  as  follows:  Two  rods,  18  ft.  long;  two 
rods,  24  ft.  long;  two  rods,  30  ft.  long;  two  rods,  34  ft.  long;  and  four 
straight  rods  running  the  entire  length  of  the  girder.  These  lengths 
all  exceed  somewhat  the  requirement  for  moment.  This  is  shown 
graphically  in  the  moment  diagram  of  Fig.  62b  where  the  resisting 
moment  of  the  beam  is  shown  by  the  stepped  line,  and  the  bending 
moment  by  the  curve.  The  necessary  length  of  rod  to  develop  a  bond 


242  WORKING    STRESSES.  [Cn.  V. 

strength  equal   to   its  full  working   stress  at   the   specified  values  is 

-L—  —  =  50  diameters.    For  the  l|-in.  rods  this  is  equal  to  50  X  If  =  69  in. 
4X75 

This  is  provided  in  all  cases  with  the  bent  rods.     For  the  four  straight 
rods  the  maximum  possible  bond  stress  will  be  found   from  eq.   (1), 

80,000 

In  this  case  C7  =  ——  =  2000  Ibs.  per  lineal  inch.    The  bond 
40 


stress=-  -  -=115  lbs/in2.      The    allowed   value  is  75  XH  or  112.5 

lbs/in2.  The  calculated  stress  is  very  slightly  above  this,  but  con- 
sidering the  prolongation  of  the  beam  beyond  the  center  of  support 
it  will  be  allowed.  Additional  strength  will  be  furnished  by  hooks  at 
the  ends  of  the  rods. 

The  effectiveness  of  the  bent  rods  in  carrying  shear  will  now  be 
determined.  A  diagram  of  shearing  stress  is  shown  in  Fig.  626  pro- 
jected on  a  line  at  45°  to  the  axis.  This  will  be  convenient  in  repre- 
senting the  effect  of  the  diagonal  bars.  These  bars,  together  with 
other  reinforcement,  must  be  sufficient  to  carry  the  shear  represented 
by  the  shaded  area,  the  concrete  carrying  the  remainder.  The  point  a 
where  the  first  rod  intersects  the  neutral  axis  is  about  11.5  ft.  from  the 

11.5 
center    and    the    unit    shearing    stress  =  16  +  (100  —  16)  —  —  =  64  lbs/in2. 

This  is  shown  by  the  ordinate  a^.  The  concrete  will  carry  30  lbs/in2, 
leaving  34  lbs/in2  to  be  carried  by  the  steel.  This  amounts  to 
34X18  =  610  Ibs.  for  each  lineal  inch  of  beam.  Considering  these  rods 
effective  over  a  distance  of  3  ft.  (the  space  between  the  first  and 
second),  the  stress  in  each  rod  is,  by  eq.  (3),  Art.  125,  equal  to  iX.7X 
610X36  =  7700  Ibs.  This  gives  a  unit  stress  of  7700/1.23  =  6270  lbs/in2, 
a  low  value.  Graphically,  the  total  amount  carried  may  be  repre- 
sented by  the  shaded  area  between  the  ordinates  1-1  and  2-2  in  Fig. 
626.  In  a  similar  manner  it  is  found  that  at  point  6,  14.5  ft.  from  the 

14.5 

center,    the    shearing    stress  =  16  +  84  X  -^-  =  77  lbs/in2,   the    concrete 

carrying  30  Ibs.  and  the  rods  47  Ibs.,  giving  a  stress  in  the  rods  of 
8650  lbs/in2.  At  c  the  shear  is  90  lbs/in2,  and  the  stress  in  the  rods 
11,500  lbs/in2,  and  at  d  the  shear  is  about  98  lbs/in2,  and  the  stress 
about  11,500  lbs/in2. 

This  analysis  indicates  that  the  bent  rods  are  sufficient  to  carry 
all  the  shear  except  to  the  left  of  a.  The  maximum  value  =  58  lbs/in2, 
thus  requiring  28  lbs/in2  to  be  carried  by  steel.  This  will  be  supplied 
in  the  form  of  stirrups.  If  f-in.  stirrups  are  used  in  a  double  loop, 
their  strength,  at  say,  12,000  lbs/in2,  is  4X12,000  X.I  1=5280  Ibs.,  and 


§  136.]  COLUMNS.  243 

5280 
the  spacing  to  carry  281bs/in2  =  —  —  —  =  10.5  in.   Inasmuch  as  stirrups 


assist  in  supporting  the  rods  and  in  binding  together  web  and  flange, 
and  as  they  add  greatly  to  the  security  and  reliability  of  the  construc- 
tion, they  will  be  used  throughout  and  spaced  12  in.  apart,  except 
along  the  center  8  ft.  where  they  will  be  spaced  2  ft.  apart. 

If  stirrups  alone  were  used  to  carry  the  shear  they  would  need  to 
be  spaced  about  4  in.  apart  near  the  end. 

In  the  design  of  web  reinforcement  it  should  be  understood  that  the 
methods  of  calculation  here  used  can  be  considered  as  only  roughly 
approximate.  They  are  based  broadly  on  theoretical  considerations 
and  the  results  of  experiment  and  lead  to  satisfactory  and  safe  designs, 
but  they  cannot  be  considered  as  being  in  any  sense  precise  methods 
or  as  representing  the  best  that  may  be  developed. 


COLUMNS. 

136.  Working  Stresses. — In  determining  the  proper  working 
stresses  for  columns  it  is  necessary  to  consider  the  question 
\mainly  with  reference  to  the  stress  in  the  concrete,  for  under 
ordinary  working  stresses  in  the  concrete  the  stress  in  the  steel 
will  be  relatively  low.  From  the  tests  and  discussion  of  pre- 
ceding chapters  it  appears  that  with  reference  to  the  behavior 
of  the  concrete,  columns  may  be  divided  into  two  classes: 
(1)  columns  reinforced  with  longitudinal  reinforcement  only, 
and  (2)  columns  reinforced  with  hoops  or  bands  and  with  or 
without  longitudinal  reinforcement.  These  types  will  be  con- 
sidered separately. 

(1)  Columns  Reinforced  ivith  Longitudinal  Steel  Only. — In 
this  form  of  column  the  concrete  fails  in  a  manner  similar  to 
the  failure  of  an  unreinforced  column.  When  a  load  is  reached 
which  stresses  the  concrete  to  about  the  same  value  as  in  a 
plain  concrete  column,  failure  takes  place  suddenly  and  by 
shearing  action.  The  ultimate  strength  of  the  entire  column  is, 
however,  increased  by  the  steel  and  approximately  as  theory 
would  indicate.  Considering  the  manner  of  failure  and  the 
lack  of  " toughness"  in  such  a  column  the  factor  of  safety 
should  be  relatively  large  and  determined  on  about  the  same 

•4**   4y     .    '  Jt.   -r^t^i     *bj*T~t,    ltd:,   &++*>.     d  £<  *^ 

,  ,t    ^      '  tf    C^'-^t^^^^f,     fat***.  4,  /4/0 . 

<7  <7  <7 


244  WORKING  STRESSES.  [Cn.  V. 

basis  as  for  a  short  column  of  plain  concrete.  The  strength  of 
a  1:2:4  concrete  at  60  days  in  the  form  of  a  short  column  or 
cylinder  ranges  from  about  1600  to  1800  lbs/in2,  and  applying 
a  factor  of  safety  of  four,  gives  a  working  stress  of  400  to  450 
lbs/in2.  This  may  be  taken  as  a  suitable  value  for  the  con- 
crete in  the  type  of  column  here  considered.  For  richer  and 
stronger  mixtures  the  working  stress  may  be  increased  accord- 
ingly. Where  the  concrete  is  depended  upon  to  fire-proof  the 
steel,  a  certain  thickness  should  be  deducted  in  calculating 
strength.  As  shown  in  Art.  141  the  necessary  thickness  for 
fire-proofing  is  about  two  inches,  but  if  1J  inches  be  deducted 
all  around  in  calculating  strength  this  will  amply  provide  for 
the  weakening  effect  of  fires.' 

The  working  stress  in  the  steel  is  a  function  of  the  working 
stress  in  the  concrete  and  the  ratio,  n,  of  the  moduli  of  elas- 
ticity of  the  two  materials.  If  this  ratio  is  taken  at  12,  then 
with  a  stress  of  400  in  the  concrete,  the  stress  in  the  steel  will 
be  12X400  =  4800  lbs/in2.  Under  working  loads  the  steel  is 
therefore  stressed  only  to  a  very  low  value. 

Let  us  consider  the  variation  in  the  stresses  in  a  column 
subjected  to  increasing  loads,  following  the  same  general 
method  of  analysis  as  for  the  beam  in  Art.  121.  Assume  a 
concrete  having  a  compresive  strength  in  the  form  of  a  short 
column  of  1600  lbs/in2,  and  assume,  further,  that  the  stress- 
strain  diagram  is  parabolic,  as  shown  in  Fig.  63a,  with  a  value 
of  E  at  the  origin  of  3,000,000  lbs/in2.  The  value  of  E  at 
other  stresses  will  be  2,625,000  at  400  Ibs.;  2,250,000  at  800 
Ibs.;  and  1,875,000  at  1200  lbs/in2.  Consider  a  column  com- 
posed of  this  concrete  and  2%  of  steel  reinforcement.  From 
Art. '^5,  Chapter  III,  the  total  load  on  a  column  is  given  by 
the  formula  P'  =  Afc[l  +  (n—l^p,  in  which  p  =  steel  ratio,  A  — 
total  area,  and  /c  =  unit  stress  on  the  concrete.  The  average 

Pr 

unit  stress  on  the  column  will  be /==~r  ==/£!  + (r&—l}jy,,  As- 
sume a  working  stress  in  the  concrete  of  400  lbs/in2.  For  this 
stress  the  value  of  n  is  30/2.625  =  11.4,  and  the  average  unit 


§  136.] 


COLUMNS. 


245 


stress  on  the  column  section  will  therefore  be  400(1  + 10.4 X. 02) 
=  483  lbs/in2.  In  Fig.  636,  let  this  be  represented  on  the  axis 
OX  by  the  distance  Oa,  which  is  conveniently  taken  as  a  unit. 
Let  the  ordinate  ab  represent  the  corresponding  unit  stress  in 
the  concrete  of  400  lbs/in2,  and  ac  the  unit  stress  in  the  steel, 
=  400X11.4  =  4560  lbs/in2.  Now  assume  the  load  to  be  in- 
creased so  as  to  cause  a  stress  of  800  lbs/in2  in  the  concrete. 


FIG.  63a. 


Y 

1600 

A 

^B 

1200 

•»•? 

/ 

800 

b 

/ 

/a 

a, 

a! 

2 

3 

4 

• 

6000 

1 

10,000 

\ 

15,000 

v? 

^ 

80,000 

\ 

c 

25,000 

\ 

30,000 

c 

35,000 

v 

FIG.  636. 

The  value  of  n  at  this  stress  =  30/2.250  =  13.33.  The  value  of 
/,=  800X13.33  =  10,670  lbs/in2.  The  average  unit  stress  is 
/=  800(1  + 12.33 X. 02)  =997  lbs/in2.  This  will  be  represented 
by  the  abscissa  Oa'  of  a  value  of  997/483  =  2.06  units.  The 
ordinates  afb'  and  aV  (to  the  heavy  lines)  =800  and  10,670 
lbs/in2,  respectively.  In  the  same  manner  continue  the  calcu- 
lations and  plot  the  curves  OB  and  0(7,  which  will  then  repre- 
sent the  variation  of  concrete  and  steel  stress  throughout  the 
entire  range  of  load  to  the  ultimate  strength  of  the  concrete. 
At  this  point  the  stress  in  the  steel  will  be  20X1600  =  32,000 


246  WORKING    STRESSES  [Cn.  V. 

lbs/in2,  and  the  average  unit  stress  will  be  1600(1  + 19  X. 02)  = 
2208  lbs/in2.  This  is  4.57  times  the  load  causing  the  stress  of 
400  lbs/in2.  From  this  it  is  plain  that  with  increasing  loads 
the  steel  receives  a  greater  proportionate  stress,  the  variation 
in  the  amount  carried  by  the  steel  depending  on  the  variation 
in  the  value  of  n.  It  is  also  evident  from  this  diagram  that 
the  ultimate  load  on  the  column  is  greater  than  four  times  the 
load  (4.57  times  in  the  assumed  case)  which  produces  the 
stress  of  400  lbs/in2  in  the  concrete.  Hence  if  the  working 
stress  in  the  concrete  is  based  on  a  factor  of  safety  of  four 
relative  to  plain  concrete,  then  the  factor  of  safety  of  the  rein- 
forced column  will  be  greater  than  four.  The  case  is  somewhat 
similar  to  that  of  the  beam.  Obviously  the  total  load  increases 
more  rapidly  than  the  value  of  the  stress  fc,  the  exact  rate 
depending  on  the  relative  amount  of  steel  and  the  variation 
in  n. 

In  order  to  secure  a  more  uniform  factor  of  safety,  and  to 
take  some  account  of  the  fact  that  under  increasing  loads  the 
steel  receives  an  increasing  proportion,  it  is  desirable  to  use  a 
value  of  n  in  the  calculations  somewhat  larger  than  that  which 
is  obtained  by  taking  a  value  of  Ec  corresponding  to  very  low 
stresses.  On  this  basis  the  actual  stress  in  the  concrete  at 
working  load  will  be  slightly  greater  than  assumed,  and  that 
in  the  steel  somewhat  less,  but  the  calculated  and  actual 
stresses  will  coincide  at  about  one-half  of  the  ultimate  load  and 
the  factor  of  safety  will  still  be  somewhat  greater  than  the  ratio 
of  ultimate  strength  of  the  concrete  to  its  assumed  working 
strength.  In  Fig.  636  the  dotted  straight  lines  AB'  and  AC' 
represent  the  assumed  variation  of  stress,  using  a  constant 
value  of  n  =  15.  The  average  unit  stress  on  the  column,  at 
working  loads,  will  be /=  400(1 +  14 X. 02)  =  512  lbs/in2.  This 
is  6%  greater  than  the  load  represented  by  the  abscissa  Oa, 
and  is  represented  by  the  distance  Oa\.  The  ordinates  to  the 
curves  OB  and  OC  show  the  actual  stresses  in  the  concrete  and 
steel.  The  ultimate  strength  of  the  column  being  2208  lbs/in2, 
the  real  factor  of  safety  =  2208/512  =  4.3.  A  value  of  15  for  n 


§  136.]  COLUMNS.  247 

may  well  be  used  for  all  ordinary  mixtures  and  for  all  types  of 
columns. 

(2)  Columns  Reinforced  with  Hoops  or  Bands  and  ivith  or 
without  Longitudinal  Steel. — From  the  tests  given  in  Chapter 
IV,  it  is  seen  that  in  general  the  effect  of  hooping  is  to  increase 
the  " toughness"  and  the  ultimate  strength  of  the  column. 
The  elastic  limit  and  rigidity  of  the  column  appears  to  be 
decreased  if  anything.  When  used  with  longitudinal  steel  the 
hooping  acts  in  much  the  same  way,  but  is  of  greater  import- 
ance in  this  case  as  it  keeps  the  concrete  intact  up  to  a  degree 
of  deformation  that  enables  the  longitudinal  steel  to  be  stressed 
to  its  elastic  limit.  It  thus  renders  such  reinforcement  very 
effective. 

Concerning  the  proper  working  stress  for  hooped  columns, 
it  would  seem  that  this  should  be  selected  mainly  with  refer- 
ence to  the  elastic  limit,  as  in  the  case  of  structural  steel;  but 
the  greater  toughness  of  the  hooped  column,  as  compared  to 
the  other  type,  insures  a  much  larger  and  more  certain  margin 
of  safety,  and  hence  the  working  stress  may  be  made  a  greater 
proportion  of  its  elastic  limit  strength  than  in  the  other  case. 
The  two  types  of  columns  may  be  compared  to  mild  steel  and 
cast  iron;  a  much  higher  relative  working  stress  may  be  used 
in  the  former  than  in  the  latter,  chiefly  because  of  its  larger 
margin  of  safety  against  deformation  beyond  the  elastic 
limit.  This  is  of  great  importance,  especially  with  respect  to 
effects  of  unequal  settlement,  eccentric  loading  and  secondary 
stresses. 

For  hooped  columns,  without  longitudinal  steel,  the  elastic 
limit  is  about  the  same  as  for  plain  concrete  and  varies  but 
little  for  various  percentages  of  steel.  Hence  the  same  working 
stress  may  be  used  for  all  percentages  of  hooping;  but  for 
reasons  already  stated  this  value  may  be  made  greater  than 
for  plain  or  for  longitudinally  reinforced  columns.  For  this 
type  of  column,  therefore,  the  authors  would  suggest  a  working 
stress  about  20%  greater  than  for  plain  concrete,  or  from  500 
to  550  lbs/in2  for  an  amount  of  hooping  of  1%  or  more.  This 


248  WORKING    STRESSES.  [Cn.  V. 

value  is  to  be  applied  to  the  concrete  alone,  and  the  hooping  is 
not  to  be  taken  directly  into  account.  For  large  amounts  of 
hooping,  somewhat  higher  stresses  might  be  used,  but  increased 
strength  and  rigidity  can  be  provided  more  effectively  by  add- 
ing longitudinal  reinforcement. 

For  hooped  columns  containing  longitudinal  reinforcement, 
the  elastic  limit  of  the  column  tends  to  approach  a  point  corre- 
sponding to  the  elastic  limit  of  the  longitudinal  steel,  the  exact 
effect  depending  upon  the  effectiveness  of  the  hooping  and  the 
amount  of  longitudinal  steel.  If  this  were  fully  accomplished, 
the  working  stress  might  be  placed  as  high  as  15,000  lbs/in2 
on  the  steel,  corresponding  to  a  stress  on  the  concrete  of  about 
1000  lbs/in2.  This  is  beyond  the  normal  elastic  limit  strength 
of  the  material,  and  is  not  to  be  recommended.  Considering 
all  the  factors  involved,  it  would  seem  that  the  stress  on  the 
concrete  could  safely  be  taken  at  50%  more  than  for  plain 
concrete.  This  would  give  a  value  of  600  to  675  lbs/in2,  with 
9000  to  10,000  lbs/in2  on  the  longitudinal  steel.  To  render 
such  stresses  safe,  an  amount  of  hooping  equal  to  1%  would 
appear  from  the  results  of  tests  to  be  sufficient.  More  hooping 
will  increase  the  ultimate  strength,  but  not  materially  the 
elastic  limit,  and  hence  it  will  not  permit  the  use  of  higher 
stresses.  For  rich  concrete  still  higher  values  may  be  used, 
but  not  to  exceed  about  800  lbs/in2  on  the  concrete,  corre- 
sponding to  12,000  lbs/in2  on  the  steel.  With  our  present 
knowledge,  also,  it  would  be  unwise  to  depend  upon  the  steel 
to  carry  its  full  share  of  stress  as  here  calculated  for  very  large 
percentages.  With  5%  of  steel  and  700  lbs/in2  in  the  concrete, 
the  average  unit  stress  on  the  column  would  be  /=  700(1  + 14  X 
.05)  =  1190  lbs/in2. 

In  the  determination  of  the  strength  of  hooped  columns, 
only  the  section  within  the  hooping  should  be  considered. 
The  shell  outside  is  of  the  same  character  as  plain  concrete  and 
it  is  found  to  crack  and  split  off  at  deformations  corresponding 
to  the  ultimate  strength  of  plain  concrete.  It  is  useful  as  fire- 
proofing,  but  its  limitations  of  deformation  is  another  reason 


§  136.]  COLUMNS.  249 

for  not  selecting  too  high  values  for  the  working  stress  on  the 
core. 

It  should  be  said  that  the  above  treatment  of  the  hooped 
column  is  quite  different  from  that  of  Considere  and  of  the 
French  Commission  on  Reinforced  Concrete.  These  authori- 
ties recommend  that  the  hooping  be  counted  upon  directly  to 
a  much  larger  extent  than  the  longitudinal  reinforcement. 
The  formula  recommended  by  the  French  Commission  is 


in  which  fc  is  the  safe  strength  of  plain  concrete,  taken  at  28% 
of  the  ultimate  strength  in  the  form  of  cubes,  p  =  ratio  of  lon- 
gitudinal reinforcement,  and  pf  —  ratio  of  spiral  reinforcement. 
It  is  also  recommended  that  the  maximum  stress  shall  not 
exceed  0.6  of  the  ultimate  strength  of  the  concrete.  These 
values  are  based  chiefly  on  a  consideration  of  ultimate  strength. 
(3)  Columns  Reinforced  by  Structural  Steel  Column  Units.— 
Where  a  large  amount  of  reinforcement  is  desired,  certain 
advantages  are  gained  by  arranging  it  in  the  form  of  struc- 
tural column  units,  such  as  four  angles  latticed  together,  which 
in  themselves  are  capable  of  acting  as  columns.*  The  con- 
struction can  be  so  arranged  that  the  steel  columns  will  carry 
the  false  work  and  dead  load  of  two  or  more  floors,  thus  enabling 
the  placing  of  concrete  to  proceed  simultaneously  on  several 
floors.  In  this  way,  also,  some  initial  dead  load  stress  can  be 
applied  to  the  steel  of  the  column  before  the  concrete  of  the 
column  is  placed,  thus  enabling  higher  steel  stresses  to  be  used. 
On  the  other  hand,  such  steel  is  much  more  costly  per  pound 
than  rods.  Furthermore,  the  results  of  experiments  show  that 
the  adhesion  of  concrete  to  steel  where  the  latter  presents  broad 
flat  surfaces  is  not  good,  and  the  presence  of  numerous  lattice 
bars  hinders  the  production  of  a  dense  homogeneous  concrete. 

*  For  a  good  example  of  such  a  design,  see  paper  by  Wm.  H.  Burr  on 
"  The  Reinforced  Concrete  Work  of  the  McGraw  Building,"  Trans.  Am. 
Soc.  C.  E.,  Vol.  GO,  1908. 


250  WORKING    STRESSES.  [Cn.  V. 

The  resulting  column  is  therefore  likely  to  be  less  of  a  mono- 
lithic character  than  one  in  which  the  reinforcement  consists 
of  small  rods.  In  order  that  the  concrete  may  be  counted 
upon  in  such  a  column,  it  should  be  well  enclosed  either  by 
the  structural  form  itself  or  by  means  of  bands  or  hooping.  All 
concrete  not  so  enclosed  can  be  considered  only  as  fire-proofing. 
Where  designed  in  accordance  with  these  principles,  and 
the  steel  and  concrete  receive  their  load  simultaneously,  the 
working  stresses  may  be  taken  about  the  same  as  for  the 
second  class  of  columns  here  discussed.  If,  however,  a  partial 
load  is  applied  to  the  steel  before  the  concrete  is  placed,  such 
initial  stress  need  not  be  counted,  excepting  that  the  total 
stress  in  the  steel  should  not  exceed  the  usual  working  stress 
for  steel  columns  of  about  16,000  lbs/in2.  Where  the  amount 
of  steel  becomes  very  large  the  relative  value  of  the  concrete 
becomes  more  uncertain  and  its  consideration  as  an  element 
of  strength  is  of  doubtful  wisdom  and  unsupported  by  experi- 
mental evidence. 

137.  Long  Columns. — The  tests  of  Chap.  IV  indicate  that 
for  lengths  of  20  to  25  diameters  little  or  no  difference  in 
strength  is  shown  for  different  lengths.     Very  long  columns 
should,   however,   be  avoided,   and  it  is  important  to  adopt 
a  conservative  practice  in  this  regard.      It  is  therefore   ad- 
visable   to   apply  the    long  column    formula    of  Art.  966  for 
lengths  exceeding   12  or   15  diameters.     It  would  also  seem 
that  a  ratio  of  I  to  r  greater  than  100  should    be  not  used, 
or  a  ratio  of  length  to  least  width  greater  than  about  30. 
Banded  columns  are  much  to  be  preferred  for  slender  pro- 
portions. 

It  is  important  to  note  that  plain  concrete  is  entirely 
suitable  for  short  columns  up  to  lengths  of  6  to  10  diameters, 
and  for  such  columns  the  addition  of  steel  is  not  in  general 
economical. 

138.  Column  Details. — In    the    construction    of  columns 
great  care  should  be  exercised  to  place  and  hold  the  steel  in 
its  proper  position  and  to  secure  sound  work.     In  this  respect 


§  138.]  COLUMNS.  251 

poor  workmanship  is  more  serious  perhaps  than  in  any  other 
structural  form.  Eccentricity  of  steel  or  uneven  quality 
of  concrete  not  only  causes  weakness  at  the  section  in  ques- 
tion, but  also  results  in  eccentricity  of  load  and  lateral 
deflections.  Reinforcing  rods  must  be  arranged  concentri- 
cally and  held  securely  in  place  until  the  concrete  is  set.  This 
is  usually  accomplished  by  wiring  or  banding  the  rods  together 
at  intervals  of  a  foot  or  so,  but  such  banding  cannot  be  con- 
sidered as  hooping  in  the  sense  usually  employed.  Where 
hooping  is  used  as  reinforcement  it  may  consist  of  wire  spirally 
wound  or  otherwise,  or  of  separate  bands  of  welded  or  riveted 
steel.  To  be  effective  such  hooping  should  be  spaced  relatively 
close,  so  as  to  serve  to  confine  the  concrete  within  the  cylinder 
formed  by  the  hooping  and  to  effect  the  "  toughening " 
assumed  in  the  previous  discussion.  What  such  spacing  may 
be  is  not  well  determined,  but  until  further  evidence  is  avail- 
able a  clear  spacing  of  about  one-fifth  to  one-fourth  the 
diameter  of  the  hoops  or  bands  may  be  considered  the  maxi- 
mum. A  total  amount  of  hooping  or  banding  at  least  equal 
to  1%  of  the  enclosed  volume  should  be  used.  The  French 
Commission  recommends  a  spacing  of  spirals  of  one-eighth 
to  one-fifth  the  column  diameter,  but  in  this  case  the  spiral 
reinforcement  is  counted  UDon  directly  in  the  calculation  of 
strength. 

•  In  the  case  of  hooped  columns  or  columns  in  which  lateral 
reinforcing  members  are  used,  such  as  lacing  on  structural 
units,  special  care  should  be  taken  to  secure  as  dense  concrete 
as  possible,  and  to  reduce  the  settlement  of  the  material  to  a 
minimum.  Any  settlement  tends  to  create  vacant  spaces  or 
porous  material  underneath  the  reinforcement.  In  splicing 
columns  large  rods  or  structural  shapes  should  be  accurately 
fitted  and  well  spliced;  small  rods  may  be  spliced  at  floor 
levels  by  overlapping  a  sufficient  distance  to  develop  the 
requisite  bond  strength.  At  the  base  of  a  column  large  rods 
or  shapes  should  rest  upon  suitable  base  plates  in  the  founda- 
tion concrete. 


252  WORKING    STRESSES.  [Cn.  V. 

139.  Economy  in  the  Use  of  Reinforced  Columns. — From 
eq.  (l),Art.  95,  we  see  that  with  a  value  of  n=15,  the  use  of 
each  1%  of  longitudinal  steel  adds  14%  to  the  strength  of  a 
column.  If  the  ratio  of  cost  of  steel  to  cost  of  concrete  per 
unit  volume  be  50,  then  the  increased  cost  of  a  column  with 
1%  of  steel  will  be  50X1%  =  50%.  The  gain  in  strength 
being  only  14%,  the  relative  economy  of  the  reinforced  column 
is  only  114/iso  =  76%  that  of  the  plain  concrete.  Again,  take  a 
very  strong  mixture,  such  as  1:1  mortar,  whose  working 
stress  may  possibly  be  .taken  as  high  as  800  lbs/in2.  Such  a 
mortar  will  cost  perhaps  $12.00  per  cu.  yd.  (not  including 
forms,  etc.)  or  45  c.  per  cu.  ft.  Placing  steel  at  the  low  value 
of  2  c.  per  lb.,  the  cost  ratio  becomes  22.5.  Such  concrete 
will  have  a  value  of  Ec  of  at  least  3,000,000,  giving  ft  =10. 
Hence  1%  reinforcement  will  add  9%  to  the  strength  and 
22.5%  to  the  cost.  If  a  cheap  concrete  be  taken  with  a  low 
modulus  the  steel  will  add  a  larger  percentage  of  strength,  but 
at  the  same  time  a  much  greater  percentage  of  cost.  Another 
way  of  considering  this  question  is  from  the  standpoint  of 
the  relative  working  stresses  in  concrete  and  steel.  Using  a 
value  of  ft  =  15  the  load  carried  by  the  steel  per  square  inch 
is  fifteen  times  that  taken  by  the  concrete.  If  the  cost  ratio 
is  50  then  the  steel  is  50/15  =  3  J  times  as  costly  as  the  con- 
crete for  the  same  service. 

The  above  analysis  shows  that  from  the  standpoint  of 
theoretical  economy  the  use  of  steel  in  columns  is  undesirable, 
and  were  this  the  only  consideration  it  would  not  be  used,  at 
least  in  the  form  discussed.  While  no  economy  can  be  figured 
for  the  use  of  steel  in  columns  it  is  by  no  means  valueless.  In 
practice,  columns  are  subjected  to  bending  moments  uncertain 
in  amount,  but  for  which  something  more  than  plain  concrete  is 
desired,  especially  where  the  column  is  of  considerable  length. 
It  is  in  such  columns  that  tensile  stresses  are  most  apt  to  occur 
and  where  steel  is  most  needed.  Furthermore,  steel  is  a  more 
reliable  material  than  concrete,  and  in  small  sections  where  the 
danger  of  weak  or  imperfect  spots  in  the  concrete  is  greatest, 


§  140.]  DURABILITY    OF    REINFORCED    CONCRETE.  253 

steel  reinforcement  is  of  great  value  in  producing  a  more  reliable 
structure.  Then,  again,  great  strength  may  be  desired  from 
small  sections  in  order  to  save  space,  in  which  case  steel  may  be 
used.  In  very  large  (relatively  short)  columns  little  is  to  be 
feared  from  bending  stresses,  as  in  such  a  case  no  resultant 
tensile  stress  is  likely  to  occur. 


DURABILITY   OF  REINFORCED  CONCRETE. 

140.  The  Protection  of  Steel  from  Corrosion. — A  continuous 
coating  of  Portland  cement  has  been  found  by  experience  to 
be  a  practically  perfect  protection  of  steel  against  corrosion. 
The  rusting  of  iron  requires  the  presence  of  moisture  and  carbon 
dioxide.  Portland  cement  not  only  forms  a  coating  which  ex- 
cludes the  moisture  and  C02,  but  in  hardening  it  absorbs  C02, 
tending  to  remove  any  of  this  gas  which  may  be  present.  In 
practice  the  protective  nature  of  Portland-cement  concrete  has 
long  been  known,  and  its  use  as  a  paint  was  adopted  by  the 
Boston  Subway  Engineers  after  careful  investigation. 

While  an  unbroken  coating  of  cement  offers  what  appears 
to  be  a  perfect  protection,  the  value  of  a  concrete  as  actually 
deposited  may  be  very  much  less.  A  series  of  experiments 
made  by  Professor  Charles  L.  Norton  gives  valuable  information 
on  this  subject.  In  one  series,  small  specimens  of  steel  6"  long 
were  embedded  in  blocks  3"X3"x8"  in  size  of  various  mix- 
tures of  cement,  sand,  and  stone  or  cinders.  The  blocks  were 
then  exposed  for  three  weeks  to  various  corrosive  atmospheres 
consisting  of  steam,  air,  and  C02.  The  results  were  as  follows: 
The  neat  cement  furnished  perfect  protection.  The  specimens 
embedded  in  mortars  and  concretes  showed  spots  of  rust  at 
voids  or  adjacent  to  a  badly  rusted  cinder.  He  concludes  that 
concrete  to  be  an  effective  protection  should  be  mixed  quite 
wet  so  as  to  furnish  a  thin  coating  on  the  metal,  and  must  be 
free  from  voids  and  cracks.  He  finds  that  dense  cinder  concrete 
mixed  wet  is  as  effective  as  stone  concrete. 

In  a  second  series  of  experiments  on  steel  already  rusted, 


254  WORKING    STRESSES.  [Cn.  V. 

from  a  slight  stain  to  a  deep  scale,  the  following  results  were 
obtained:  The  concrete  was  1 :  2| :  5  (stone)  and  1:3:6 
(cinders).  After  one  to  three  months  in  corroders  and  one  to 
nine  months  in  damp  air  no  specimen  showed  any  change 
except  where  the  concrete  was  poorly  applied.  Some  of  the 
concrete  was  purposely  made  very  dry  and  the  rods  were  not 
well  covered.  These  specimens  were  seriously  corroded.  Un- 
protected steel  specimens  subjected  to  the  same  treatment  were 
almost  entirely  corroded.  While  the  experiments  of  Professor 
Norton  provided  for  a  covering  of  1J  inches,  there  is  no  reason 
to  suppose  that  a  much  thinner  covering,  if  intact,  v.  ill  not 
furnish  as  good  protection. 

Many  cases  have  been  cited  of  steel  removed  from  concrete 
after  the  lapse  of  20  years  or  more  and  found  to  be  in  perfect 
condition.  A  test  by  Mr.  H.  C.  Turner,*  in  which  steel  bars 
embedded  to  a  depth  of  3  inches  in  blocks  of  1:2:4  and  1:3:5 
concrete  and  exposed  to  sea-water  and  air  for  nine  months 
showed  perfect  preservation. 

Perfect  protection  of  the  steel  by  concrete  was  demon- 
strated in  the  case  of  a  building  at  New  Brighton,  N.  Y.,  built 
in  1902  and  partly  torn  down  in  1908.  All  steel  was  found 
to  be  in  perfect  condition  excepting  in  a  few  cases  where 
column  hoops  came  closer  than  f  inch  to  the  surface.  The 
footings  were  covered  by  the  tide  twice  daily  but  the  bars 
therein  showed  no  corrosion. f 

In  view  of  such  tests  and  observations  as  here  noted  it 
may  be  concluded  that  when  well  placed  the  concrete  affords 
complete  protection  against  corrosion. 

141.  Fireproofing  Effect  of  Concrete. — Severe  fire  tests 
show  that  when  concrete  is  subjected  to  red-hot  temperatures 
(about  1700°)  for  three  or  four  hours  and  then  is  quenched  by 
hose  streams,  it  is  likely  to  show  pitting  but  that  it  will  still 
offer  a  sufficient  protection  to  the  steel.  J 

*  Eng.  News,  Aug.  1904,  p.  153. 
f  Eng.  Record,  Vol.  57,  1908,  p.  105. 

J  See  tests  by  Professor  Ira  W.  Winslow  in  Eng.  Record,  Nov.  26,  1904, 
p.  634,  and  by  Professor  F.  P.  McKibben  in  Eng.  News,  Nov.  21,  1901,  p.  378. 


§  141.]  FIREPKOOFING    EFFECT    OF   CONCRETE.  255 

A  reinforced-concrete  building  at  Bayonne,  N.  J.,  was  sub- 
jected to  a  very  hot  fire  in  the  burning  up  of  its  contents  but 
with  no  injury  to  the  building.* 

In  the  Baltimore  fire  of  1904  the  value  of  concrete  as  a 
fireproofing  material,  and  of  reinforced-concrete  construction, 
was  fully  demonstrated.  Professor  C.  L.  Norton  of  the  Insur- 
ance Engineering  Experiment  Station,  after  a  careful  study  of 
the  damage  done  by  the  fire,  states  as  follows:! 

"Where  concrete  floor  arches  and  concrete-steel  construc- 
tion receive  the  full  force  of  the  fire  it  appears  to  have  stood 
well,  distinctly  better  than  the  terra-cotta."  The  reason  for 
this  he  considers  to  be  the  fact  that  terra-cotta  expands  about 
twice  as  much  as  steel,  but  that  concrete  expands  about  the 
same.  Little  difference  was  observed  between  stone  and 
cinder  concrete.  High  temperatures  long  continued  dehydrate 
and  soften  concrete,  but  this  process  in  itself  gives  off  water  and 
absorbs  the  heat,  thus  protecting  the  interior.  The  layer  of 
changed  material  is  then  a  better  non-conductor  than  before, 
so  the  process  goes  on  very  slowly.  Captain  J.  S.  Sewell,  report- 
ing to  the  Chief  of  Engineers  J  on  the  Baltimore  fire,  states 
that,  with  reference  to  concrete  construction  subjected  to  very 
high  heats:  "Exposed  corners  of  columns  and  girders  were 
cracked  and  spalled,  showing  a  tendency  to  round  off  to  a  curve 
of  about  3  in.  radius.  Where  the  heat  was  most  intense  the  con- 
crete was  calcined  to  a  depth  of  J"-£",  but  showed  no  tendency 
to  spall,  except  at  exposed  corners.  On  wide,  flat  surfaces 
the  calcined  material  was  not  more  than  J-in.  thick  and  showed 
no  disposition  to  come  off.  The  terra-cotta  fireproofing  showed 
up  much  poorer."  In  his  general  conclusions  he  considers  it 
at  least  as  desirable  as  steel  work  protected  by  the  best  com- 
mercial hollow  tiles,  and  preferable  to  tile  for  floor  slabs  and 
fire-proof  covering. 


*  Eng.  Record,  April  12,  1902,  p.  341. 
f  Eng.  News,  June  2,  1904,  p.  524. 
t  Eng.  News,  March  24,  1904. 


256  WORKING    STRESSES.  [Cn.  V. 

In  a  report  of  a  committee  of  members  of  the  American 
Society  of  Civil  Engineers  on  the  effects  of  fire  in  the  San 
Francisco  conflagration,  similar  conclusions  were  reached  as 
to  the  value  of  concrete  as  a  fireproofing  material.  It  was 
also  found  far  preferable  to  tile  for  floors.  With  respect  to 
the  injury  to  the  concrete  itself  the  committee  was  of  the 
opinion  that  it  was  sufficient  in  many  cases  to  require  recon- 
struction.* Additional  evidence  of  the  value  of  concrete  as 
a  fireproofing  material  is  contained  in  a  report  of  a  com- 
mittee of  the  National  Fire  Protection  Association,  f  This 
report  also  goes  far  to  indicate  the  necessary  thickness  of  the 
protective  covering. 

The  necessary  thickness  of  concrete  to  furnish  adequate 
fire  protection  depends  somewhat  upon  the  character  and  im- 
portance of  the  member.  Such  members  as  main  girders,  where 
a  failure  would  involve  a  considerable  portion  of  the  building 
and  where  the  steel  is  concentrated  in  a  few  rods,  should  be 
more  thoroughly  protected  than  floor  slabs  of  small  span,  where 
a  few  local  failures  would  be  of  no  importance,  and  where  addi- 
tional covering  would  add  largely  to  the  expense.  Results  of 
fire  tests  and  experience  in  conflagrations  indicate  that  2"-2J" 
will  offer  practically  complete  protection,  and  that  a  minimum 
of  \"-\"  for  floor  slabs  will  usually  be  sufficient.  Large  flat 
surfaces,  such  as  floor  slabs,  are  less  exposed  than  the  corners 
of  projecting  forms  like  beams  and  columns. 

While  satisfactory  protection  of  the  steel  can  thus  be 
secured  the  effect  of  fire  upon  the  concrete  itself,  and  its  use- 
fulness after  more  or  less  calcination,  is  a  question  of  much 
importance.  Where  a  sufficient  allowance  has  been  made 
for  such  damaged  material  it  would  appear  that  the  removal 
of  the  soft  or  loosened  portions  and  replastering  by  cement 
mortar  would  generally  secure  effective  repair. 


*  Trans.  Am.  Soc.  C.  E.,  Vol.  59,  1907. 
f  Eng.  News,  Vol.  59,  1908,  p.  627. 


§  142.]        SHRINKAGE   AND  TEMPERATURE    STRESSES.  257 

142.  Reinforcing  Against  Shrinkage  and  Temperature 
Stresses. — Where  a  reinforced  structure  is  unrestrained  by 
outside  forces  the  only  stresses  arising  from  shrinkage  and 
temperature  changes  are  those  due  to  the  mutual  action  of  steel 
and  concrete.  As  the  two  materials  have  nearly  equal  rates  of 
expansion  temperature  changes  will  cause  very  little  stress. 
Shrinkage  in  hardening  will  cause  more  important  stresses,  as 
shown  in  Art.  43,  but  still  not  unduly  large  unless  the  steel 
ratio  is  very  high. 

When  the  structure  is  restrained  by  outside  forces  so  that 
it  is  not  free  to  contract  or  expand,  as  in  the  case  of  a  long  wall, 
then  the  resulting  stresses  are  likely  to  be  high.  When  not 
reinforced,  concrete  will,  under  such  circumstances,  crack  at 
intervals,  its  maximum  deformation  under  stress  not  being 
equal  to  its  maximum  temperature  deformations.  If  it  be 
assumed  that  concrete  when  reinforced  will  not  stretch  more 
than  plain  concrete,  as  seems  probable  (Art.  42),  then  no  amount 
of  reinforcement  can  entirely  prevent  contraction  cracks.  The 
reinforcement  can,  however,  force  such  cracks  to  take  place 
as  they  do  in  a  beam — at  such  frequent  intervals  that  the 
requisite  deformation  takes  place  without  any  one  crack  be- 
coming large.  Laboratory  tests  on  beams  would  indicate  that  if 
steel  is  used  in  sufficient  quantities  the  cracks  may  easily  remain 
quite  invisible  and  be  of  no  consequence  from  any  practical 
standpoint.  Thus  if  the  coefficient  of  expansion  be  .000006 
a  change  of  temperature  of  50°  causes  a  change  of  length 
(if  free)  of  .0003  part.  A  deformation  of  this  amount  in  a 
beam  (corresponding  to  a  steel  stress  of  9000  lbs/in2)  would 
not  cause  cracks  easily  detected.  The  prevention  of  large 
cracks  by  means  of  reinforcement  is  then  a  matter  of  using 
sufficient  steel  to  force  the  concrete  to  crack  at  small  intervals. 
No  one  crack  will  open  up  far  until  the  steel  is  stressed  beyond 
its  elastic  limit,  hence  we  may  say  approximately  that  the 
amount  of  steel  used  must  be  such  that  the  concrete  will  crack 
elsewhere  before  the  steel  is  stressed  beyond  its  elastic  limit. 
A  larger  amount  of  steel  will  serve  to  keep  the  cracks  smaller. 


258  WORKING    STRESSES.  [Cn.  V. 

The  size  and  distribution  of  the  cracks  will  also  depend 
upon  the  bend  strength  furnished  by  the  rods.  If  we  assume 
the  cracks  to  develop  successively  the  distance  between  cracks 
must  be  sufficient  to  develop  a  bond  strength  equal  to  the 
tensile  strength  of  the  concrete.  Hence,  in  general,  the  size 
and  spacing  of  the  cracks  will  vary  inversely  with  the  bond 
strength  of  the  reinforcing  steel  per  unit  of  concrete  section. 

In  calculating  the  requiste  amount  of  steel  the  temperature 
stress  in  the  steel  itself  must  be  considered.  This  will  add  to 
its  skrinkage  stress,  so  that  its  total  stress  will  equal  its  tempera- 
ture stress  plus  the  stress  necessary  to  crack  the  concrete.  If, 
for  example,  the  assumed  drop  in  temperature  be  50°  the  tem- 
perature stress  in  the  steel  =  50  X  .0000065X30,000,000  =  9750 
lbs/in2.  If  the  tensile  strength  of  the  concrete  be  200  lbs/in2 
and  the  assumed  allowed  stress  (elastic  limit)  in  the  steel  be 
40,000  lbs/in2,  then  the  stress  available  =40,000  -9750  =30,250 

200 
lbs/in2,  and    the    required    percentage   of   steel  =  p= 


.0066.  If  the  elastic  limit  be  60,000  lbs/in2  the  steel  ratio  = 
i»  200 

P  =  60  QQQ  -  9750  =  >QQ4'  ^°r  the  PurP°ses  nere  considered 
obviously  a  high  elastic-limit  steel  is  desirable,  and  in  order 
to  distribute  the  deformation  as  much  as  possible  a  mechanical 
bond  is  advantageous. 


1^4.]  RECTANGULAR    BEAMS.  261 


?. — Plates  I-IV,  pp.  274-277,  are  diagrams  of  values 
of  k  and  y  for  various  values  of  p-}  and  values  of  R8  and  Rc 
(called  sinlply  R)  for  various  values  of  p  and  of  }8  and  fc.  The 
value  of  n  is  taken  at  10,  12,  15,  and  18  respectively. 

The  use  of  the  diagrams  in  finding  moments  of  resistance 
(Eqs.  (3)  and  (4))  and  in  determining  cross-sections  (Eqs.  (8) 
and  (9))  is  obvious.  The  proper  steel  ratio,  p,  to  use  for 
given  values  of  fs  and  fc  (Eq.  (7))  is  determined  from  the 
intersection  of  the  curves  for  the  given  values  of  f8  and  /c. 
Finally,  the  actual  fibre  stress,  fs  or  /c,  resulting  from  a  given 
M,  p,  and  bd2  will  be  found  by  first  calculating  M/bd2  from 
the  given  values.  Call  this  R.  Then  with  this  value  of  K  and 
the  given  value  of  p  enter  the  diagram  and  find  the  corre- 
sponding values  of  /,  and  fc. 

ILLUSTRATIVE  EXAMPLES.  —  1.  Moment  of  Resistance.  —  Given  the 
following:  6=-12",  d=20",  /*  =  14,000,  /c=600,  and  p=0.8%;  find  M, 
and  Mc.  Assume  n  =  15.  Solution.  From  PlateJtII,  p.  276,  we  find  for 
p=0.8%  and  /«  =  14,000,  fls=96;  and  for  /c=600,  #c  =  100.  Hence 
M8  =966^  =460,800  in-lbs.,  and  MC  =  100W=  480,000  in-lbs. 

2.  Fibre  Stresses.— Given  6  =  12",  d=2Q",  p=0.8%,  and  M  =450,000 
in-lbs.,  to  find  fs  and  fc.    Solution.  Use  Eqs.  (5)  and  (6)  directly;    or, 
find  M/bd2  and  use  the  diagrams.     Thus  M/bd2  =450,000/4800  =93.75. 
Then  from  Plate  III,  with  72=93.75  and  p=0.8%  we  find  /8=about 
13,500  and  /c=about  560  lbs/in2. 

3.  Cross-section  of  Beam  and  Steel  Ratio. — Given  M  =500,000  in-lbs., 
fs  =  12,000,  /c=500,  to  find  bd*.    Solution.  From  Plate  III  we  find  at 
the  intersection  of  the  curves  for  fa  =  12,000  and  /c=500,  a  value  of 
R  of  84.     Hence  bd?  =  500,000/84  =5950.    The  required  amount  of  steel 
is  also  found  from  the  diagram  to  be  0.8%. 

144.  Rectangular  Beams;  Parabolic  Variation  of  Stress ; 
for  Ultimate  Loads. 

Notation. — 

As  in  Art.  143,  but  here  Rc  =  %fJc]> 
Formidas. 

Position  of  neutral  axis, 

(10) 


282  FORMULAS,  DIAGRAMS,  AND    TABLES.  [Cn.  VI. 

Arm  of  resisting  couple, 


y  =  l~P  ........     (11) 

Moment  of  resistance, 

M8=fspj-bd2=R8-bd2,    .....  (12) 

Mc  =  ifckj.bd2=Rc-bd2  .....  (13) 

Approximately, 

M8=f8A.Q.8d,       ......  (12') 

Mc=fc-0.28bd2.     .^  .....  (13') 

A          fc 


Fibre  stresses, 


FIG.  65. 


T    M+jd 
'*~A~     A     ' 


Steel  ratio, 


bkd        bkd 
1 


(14) 
(15) 

(16) 


Cross-section  of  beam  for  given  bending  moment  M, 


(17) 


(18) 


§  145.] 


T-BEAMS. 


263 


Diagrams. — Plate  V,  p.  279,  is  a  diagram  of  values  of  k  and 
j  for  various  values  of  p\  and  values  of  Rs  and  Rc  for  various 
values  of  p,  /«,  and  /c.  The  full  lines  are  drawn  for  n  =  15;  the 
dotted  lines  for  /i=12.  The  fibre  stresses  are  here  assumed  as 
representing  ultimate  strengths,  and  the  diagram  is  supposed 
to  give  results  pertaining  to  ultimate  strength.  To  use  it 
for  purposes  of  designing,  the  given  loads  or  moments  should 
be  multiplied  by  the  selected  factor  of  safety,  or  the  value 
of  R  obtained  from  the  diagram  divided  by  such  factor  of  safety. 

145.  T-Beams ;  Linear  Variation  of  Stress. 


Notation.     (In  addition  to  that  of  Art.  143.) 
6=  width  of  flange; 
5'  =  width  of  web; 
t=  thickness  of  flange; 
0  =  depth  cf  resultant  of  compressive  stress; 
d—  z  =  arm  of  resisting  couple. 

Formulas. 

Case  I.  Neutral  axis  in  the  flange. 

Use  formulas  (l)-(9)  as  for  rectangular  beams, 
formula  (1)  for  k  will  determine  whether  the  case 
is  I  or  II. 


Approximately, 


(19) 


(20) 


.264  FORMULAS,  DIAGRAMS,  AND  TABLES.  CH.  VI, 

Case  II.    Neutral  axis  in  the  web;    compression  in  web 
neglected. 

Position  of  neutral  axis, 
t 


<t- 


, 

nA  +  bt  ,  f 

pn+— 

Position  of  resultant  of  compressive  stress, 

3*-4 

•--^;i  ......    ,    (22) 

2k--    3 

Moment  of  resistance, 

z),  .    ......     (23) 


z)  .....     (24) 
Steel  area, 


Assume     (d  —  2)  =  Jc?. 

Diagrams.  -Values  of  k  and  /,  for  various  values  of  p  and 
tldy  are  given  in  Plate  VI.  Plates  VII  to  XI  give  values  of 
M/bd2  from  eq.  (24)  for  various  values  of/s,/c,  and  t/d. 

146.  Beams  Reinforced  for  Compression. 

Notation.     (In  addition  to  that  of  Art.  143.) 
Ar  =  area  of  compressive  steel; 
p'=  steel  ratio  of  compressive  steel; 
//  =  unit  stress  in          '  ' 
C'  =  total  stress  in  the  compressive  steel; 
df  =  distance  from  compressive  face  to  the  plane  of  the  com- 

pressive steel; 
#=depth  to  resultant  compression,  C+C". 


§  146.]  BEAMS    REINFORCED    FOR    COMPRESSION.  265 


Formulas. 

Position  of  neutral  axis, 


-n(p+p%^.     < 


Position  of  resultant  of  compressive  stress,  C4-G", 

ft    d'6'                            2  ,Jk_ti\ 
x= -rir-d;  in  which  77= -^ .  .    .     (27) 


d     ; 


c' 


FIG.  67. 
Arm  of  resisting  couple, 

>• 

Moment  of  resistance, 


Fibre  stresses, 


M+jd 
A    ' 

ft 


(28) 


(29) 

-d').   .    .    .    (30) 


(3D 

(32) 


266 


FORMULAS,  DIAGRAMS,  AND  TABLES. 


[CH.  VI. 


Diagrams. — Values  of  k  and  j  are  given  in  Fig.  29,  p.  94, 
for  various  values  of  -p  and  of  p' .  It  is  assumed  that  d'  jd 
=  1/10  and  n  =  15.  Plate  XII,  p.  285,  gives  the  amount  of  com- 
pressive  steel  (values  of  pf)  necessary  to  use  in  order  to  reduce 
the  compressive  fibre  stress,  /c,  any  given  percentage  below 
the  value  it  would  have  with  no  compressive  reinforcement. 
The  effect  of  this  compressive  steel  upon  the  value  of  the  ten- 
sile stress  in  the  steel  is  also  given  in  the  diagram  for  various 
values  of  p  and  p' . 

147.  Flexure  and  Direct  Stress.— There  are  two  cases: 
I.  Where   there  is   compression   on   the   entire   cross-section 

(Figs.  68  and  69); 
II.  Where  there  is  some  tension  on  the  cross-section  (Fig.  70). 

fc 


FIG.  68. 


FIG.  69. 


70. 


§147]  FLEXURE    AND    DIRECT    STRESS.  267 

J 

Notation. — The  lower  side  of  the  beam  in  the  figures  on  the 
preceding  page  is    called  the'  "tension  face". 

R  =  resultant  force  acting  on  the  section; 

N  =  component  of  R  normal  to  section;    *  * 

e  =  eccentric  distance  of  R,  e/h= eccentricity; 

M = bending  moment  =  Ne ; 

A' = area  of  steel  near  compressive  face; 

p'=A'/bh; 

A  =area  of  steel  near  tension  face; 

'  p=A/bh; 

d!  =  distance  of  compressive  steel  from  face; 

u= distance  from   compressive  face  to  centroid  of  trans- 
formed section  ] 

.  a = distance  from   steel 'to  center  of  section  for  symmet- 
rical reinforcement; 

At  =  area,  of  transformed  section; 

fe=  moment   of  inertia  of  concrete  about  central  axis  of 
transformed  section; 

I8=  moment  of  inertia  of  steel  about  central  axis  of  trans- 
formed section; 

/j=moment  of  inertia  of  transformed  section; 

fe=  maximum  compressive  fibre  stress  in  concrete; 

//=  maximum  tensile  fibre  stress  in  concrete; 

ft  =  stress  in  steel  near  compressive  face; 

/,=  stress  in  steel  near  tension  face; 


Formulas. 
General. 


At=bh+n(A+A') (34) 

It=Ie+nl9,     *.....„..    (35) 
%h+npd+np'd' 


268  FORMULAS,  DIAGRAMS,  AND  TABLES.  [Cn.  VL 

Caise  I.  Compression  on  the  entire  cross-section. 

Fibre  stresses: 

N     Mu 

/C=T-+-J-,     ./VV--V-*   .    .    ;:  (37) 
A-t     it 

N     M(h  -u) 
c'=-r-      ^7—  ;,      .    .....    (38) 

•At  't 

N     Mu  -d')\ 

J):  '  '  •  :  •  (39) 

N     M(d- 

(40) 


If  //  is  negative,  then  the  case  is  Case  II. 
For  symmetrical  reinforcement  and  for  d'/d  =  l/W, 
M        1   /  a2\ 


(42) 


Case  II.  Tension  on  part  of  the  cross-section. 

If  the  tension  in  the  concrete  is  considered,  use  the  formu- 
las of  Case  I. 

For  symmetrical  reinforcement  and  for  d'/d  = 


If  the  tension  in  the  concrete  is   neglected. 
For  symmetrical  reinforcement  and  for  d' 
M       1  2pn  a2 


•  •  •  •  (47) 


§  148.]  SHEARING  AND  BOND    STRESS.  269 

Diagrams. — Values  of  I/A;  for  Case  I,  Eqs.  (41)  and  (42),  and 
Case  II,  Eqs.  (43)  and  (44),  are  given  in  Fig.  33,  p.  103;  and 
values  of  k  for  Case  II,  Eqs.  (45)  and  (46),  are  given  in  Fig.  35, 
p.  106.  Plate  XIII,  p.  287,  is  a  diagram  for  values  of  M/bh% 
for  Case  I,  Eq.  (41) ;  and  Plate  XIV  for  the  same  quantity 
for  Case  II,  Eq.  (45),  given  in  terms  of  the  eccentricity  efh 
and  the  steel  ratio  p.  The  diagrams  are  constructed  for  ft  =  15. 

ILLUSTRATIVE  EXAMPLES. — I.  An  arch  ring  is  24  in.  deep  and  is 
symmetrically  reinforced.  For  each  side  p=0.9%.  On  a  width  of 
12  in.  N=  75,000  Ibs.;  e=3  in.;  what  is  the  maximum  stress  /c? 
Solution.  The  eccentricity  =3/24  =  .125.  The  diagram  of  Plate  VII 
will  be  used,  and  the  case  is  Case  I.  This  diagram  gives  at  once 
M/bh%  =  .W7.  We  also  have  M  =75,000x3=225,000  in-lbs.  Hence 
/c  =225,0007(12  X242  X  .097)  =336  lbs/in2. 

2.  If,  in  Ex.  1,  the  eccentricity  be  6  in.,  find  the  maximum  compres- 
sive  stress  fc  and  the  maximum  tensile  stress  /c',  the  concrete  being  con- 
sidered as  carrying  tension  if  necessary.     Solution.  Use  Plate  XIII.    The 
eccentricity  is  6/24  =  .25.     From  the  diagram  we  find  Af/6/i2/c  =  .141, 
whence  /c=572    lbs/in2.      From    Eq.  (10),  p.  102,  the  value  of  fc  =  .9. 
This  being  less  than  unity  there  will  be  tension  on  the  section.     From 
Eq.  (43)  the  tensile  concrete  stress  =//= 64  lbs/in2. 

3.  If  in  Ex.  2  the  tension  in  the  concrete  be  neglected,  find  fc,  f&,  and 
/«.    Solution.     Use  Plate  XIV.     e/n  =  .25.    The  value  of  M/bh?/c  =  .U, 
whence  /c=576  lbs/in2.      The  compressive  stress  in  the  steel,  /«',  is 
always  less  than  n/c ;    in  this  case  it  is,  from  Eq.  (46),  equal  to  nfc  X 

(l—  — -)  =nfc  X  .88,  k  being  found  from  Fig.  35.     The  tensile  steel  stress, 
\       1 1/c/ 

/«,  is  less  than  the  compressive.     From  Eq.  (47)  it  is  found  to  be  276 
lbs/in2. 

148.  Shearing  and  Bond  Stress. 

Notation. 

V  =  total  vertical  shear  at  any  section; 
v  =  maximum  horizontal  or  vertical  shearing  stress  per 

unit  area; 

i/  =  average  shearing  stress  per  unit  area; 
t7=bond  stress  per  unit  length  of  beam; 
b  and  d  =  dimensions  of  a  rectangular  beam; 
b'  =  width  of  web  of  T-beam; 


270  FORMULAS,  DIAGRAMS,  AND  TABLES.  [Cn.  VI. 

d  =  net  depth  of  T-beam; 
t  =  flange  thickness  of  T-beam; 
jd  =  arm  of  resisting  couple  for  any  beam. 
Formulas. 

Rectangular  beams : 


-bjd <4») 

u  - 

Approximately, 

v  =  rij=W (49') 


„    87 
V-J-3 

T-beams: 

«=Ud< 
V 


Approximately, 


V]d' <52) 

Id (S3) 


149*  Columns. 

Notation. 

A=  total  cross-section; 

Ac= cross-section  of  concrete; 

-4«=         "  r<    longitudinal  steel; 


§  149.]  STRESSES  IN  CIRCULAR  PLATES.  271 

P-A./A; 

P  =  strength  of  plain  concrete  column; 
P'=      "          li  reinforced  column; 

/c  =  unit  stress  in  concrete; 

/«  =    ll       "       "    steel  (not  exceeding  its  elastic  limit)  ; 
fei  =  elastic-limit  strength  of  steel  ; 

/  =  average  unit  stress  for  entire  cross-section; 
pf  =  steel  ratio  of  the  hoops  of  hooped  columns. 

Formulas. 

For  short  columns;  ratio  of  length  to  least  width  not  ex- 
ceeding 20: 

/.  =  */«   ..........     (54) 

P*=fcAc+f.A.,     .......     (55) 

P'=fcA[l  +  (n-I)pl     .....     (56) 

-l)p.       ......     (57) 


If  nfc  is  greater  than  the  elastic-limit  strength  of  the 
steel,  then 

(58) 


French  Commission's  formula  for  hooped  columns: 

P'=/cA(l  +  15p  +  32pO  .....     (59) 
For  long  columns  : 


/-      Hf-Ttf, (60) 

+  20,000  W 

Diagrams.  —  Plate  XV  is  a  diagram  of  the  function 
l  +  (n-I)p  (=f/fc)  of  Eqs.  (56)  and  (57)  for  various  values 
of  p  and  values  of  n  equal  to  10,  12,  15,  20,  and  25.  The  aver- 
age working  stress,  /,  for  any  column  is  then  found  by  mul- 
tiplying the  corresponding  ordinate  from  this  diagram  by  the 
selected  working  stress  fc. 


272  DIAGRAMS,  FORMULAS,  AND  TABLES.  [On.  VL 

150.  Stresses  in  Circular  Plates.— The  exact  determina- 
tion of  stresses  in  floor  systems,  such  as  the  "mushroom"  sys- 
tem described  in  Art.  168,  and  in  the  ordinary  foundation- 
plate  supporting  a  single  column,  involves  very  complex 
analytical  processes.  As  an  aid  in  estimating  the  stresses  in 
such  cases,  Plates  XVI  and  XVII  have  been  prepared.  They 
give  the  bending  moments  in  circular  plates  supported  rigidly 
over  any  given  area  at  the  center.  Plate  XVI  gives  the  moments 
for  the  case  of  a  uniformly  distributed  load  on  the  entire  area, 
and  Plate  XVII  the  moments  for  a  load  uniformly  distributed 
along  the  periphery.  In  each  case  the  full  lines  give  the 
coefficients  for  the  radial  bending  moments,  and  the  dotted 
lines  those  for  the  circumferential  bending  moments.  The 
curves  are  drawn  for  five  different  ratios  of  rx  to  ro,  or  radius 
of  plate  to  radius  of  fixed  support.  For  other  ratios  interpo- 
lations may  be  made. 

The  calculations  for  the  diagrams  are  based  upon  the 
analysis  presente  1  by  Prof.  H.  T.  Eddy  *  for  homogeneous 
plates.  The  value  of  Poission's  ratio  assumed  in  the  numerical 
substitutions  has  been  0.1,  as  approximately  determined  in 
recent  experiments  by  Prof.  A.  N.  Talbot. 

Example. — A  circular  plate  10  ft.  in  diameter  is  rigidly  supported 
by  a  column  24  in.  in  diameter.  It  supports  a  load  of  150  lbs/ft2  over 
the  area  and  a  load  of  500  lbs/ft  along  its  outer  circumference.  Re- 
quired, the  radial  and  circumferential  bending  moments. 

Solution.  The  ratio  of  ri\r0  =  120:  24  =  5.  (The  upper  diagram  of 
Plate  XVII  may  be  used  in  finding  this  ratio.)  In  Plate  XVI  we  then  ob- 
tain the  coefficients  Q{  and  Q2  for  any  desired  point  in  the  plate,  using  the 
curves  corresponding  to  rl  -f-r0  =  5.  The  value  of  Ql  (ordinate  to  dotted 
curve)  is  seen  to  be  a  maximum  at  a  distance  from  the  center  equal  to 
about  1.7r  ;  its  value  is  about  4.7.  Hence  the  maximum  circumferen- 
tial moment  due  to  the  load  of  150  lbs/ft  is  4.7/150  ^  V  -  705  ft-lbs 
per  foot  width  of  section.  The  value  of  Q2  (ordinate  to  full  curve)  is  a 
maximum  at  the  edge  of  the  support  and  has  a  value  of  16.  The  radial 
bending  moment  is  therefore  equal  to  16  v  150 X  I2  =  2400  ft-lbs  per  foot 

*Year  Book,  Engrs.  Soc.,  Univ.  of  Minn.,  1899. 


§  152.]  TABLES.  273 

width  of  section.     The  radial  moment  rapidly  falls  off  with  increased 
distance  from  the  support. 

The  moments  due  to  the  peripheral  load  of  500  Ibs/ft  are  found  from 
Plate  XVII  to  be  respectively  M,  =  3. 1x500x1  =  1550  ft.-lbs.,  and 
M 2 = 9.6  X  500  x  1  =  4800  ft.-lbs. 

151.  Coefficients  and  Working  Stresses.— The  following  is 
a  resume  of  the  coefficients    and  working  stresses  suggested 
in  the  discussion  of  Chapter  V.     They  may  be  considered  as 
applicable   to  ordinary  conditions  on  the  basis  of  equivalent 
dead -load  stresses  and  with  concrete  of  1:2:4  to   l:2J:5  com- 
position. 

Beams. 

Working  Stress. 

Concrete  in  compression 550-650         lbs/in2 

Concrete  in  shear,  average  stress : 

a.  Without  shear  reinforcement.         30-40  " 

b.  With  shear  reinforcement...         60-100  " 

Bond  stress: 

a.  Smooth  rods 60-80  " 

6.  Deformed  rods 100-175  " 

Steel  in  tension 14,000-16,000  " 

Value  of  n=Es/Ec 12-15 

Columns. 

Concrete  in  compression 400-500 

Value  of  n=Es/Ec 15 

152.  Tables.— Areas   of  Steel   Rods—  Table   No.  19   gives 
sectional  areas  and  weights  per  foot  of  round  and  square  rods 
of  various  sizes,  and  the  total  area  per  foot  of  width  of  slab 
when  the  rods  are  spaced  various  distances  apart. 

Materials  Required  for  One.  Cubic  Yard  of  Concrete. — Table 
No.  20  gives  the  quantities  of  material  required  for  one  cubic 
yard  of  concrete  of  various  proportions.  The  table  is  based 


274  FORMULAS,  DIAGRAMS    AND    TABLES.  [Cn.  VI. 

on  Thatcher's  Tables*    As  conditions  vary  greatly,  these  tables 
should  be  used  only  for  approximate  values. 

Safe  Loads  for  Floors. — Table  No.  21  gives  span  lengths  for 
floor-slabs  for  various  live  loads  per  square  foot,  and  for  various 
values  of  working  stresses  fs  and  fc.  The  tables  have  been 
calculated  for  bending  moments  equal  to  \wl2  and  also  ^wl2. 
The  value  of  n  has  been  taken  at  15.  For  continuous  slabs 
•i-swl2  may  generally  be  taken  as  the  bending  moment.  The 
table  also  gives  the  amount  of  steel  required  per  foot  of  slab, 
so  that  by  reference  to  Table  No.  19  a  suitable  size  and 
spacing  can  readily  be  determined.  The  moment  of  resist- 
ance of  a  beam  one  foot  wide  is  also  given  for  general  use. 

*  Johnson's  Materials  of  Construction,  p.  610a. 


DIAGRAMS. 

n=io 


PLATE  1. — Coefficients  of  Resistance  of  Beams. 


276 


FORMULAS,   DIAGRAMS,    AND    TABLES. 


H.  VI. 


PLATE  II. — Coefficients  of  Resistance  of  Beams. 


110  1 

Percentage  Reinforcement 


PLATE  III. — Coefficients  of  Resistance  of  Beams. 


278 


FORMULAS,  DIAGRAMS,    AND    TABLES. 


[On.  VI. 


Percentage  Reinforcement 
05  lib     T      F  15 


PLATE  IV. — Coefficients  of  Resistance  of  Beams. 


§  152.]  DIAGRAMS. 

Full  lines  for  n=  15 ;  dotted  lines  for  n=  12, 


279 


llO  15 

Percentage  Re  nforcement 


PLATE  V.— Coefficients  of  Resistance  of  Beams. 


280 


FORMULAS,     DIAGRAMS,  AND   TABLES.  [On.  VI. 


PLATE  VI. — Values  of  k  and  /  for  T-beams. 


DIAGRAMS. 


no 
100 

90 
80 
70 


50 


X 


X 


100 
90 


_60 


50 


40 


300 


40 


10 


00 


01 


02 


03 


WluW  4  and 


04 


05 


PLATE  VII. — Coefficients  of  Resistance  of  T-beams. 


282 


FORMULAS,  DIAGRAMS,  AND  TABLES. 


[Cn.  VI. 


00 


1.00 


01 


Vahifes  of  -ft 


03 


04 


05 


1.00 


.98 


.98 


.96 


.96 


3± 


=200 


.94 


.92 


.90 


.90 


.84 


.82 


.82 


.80 


150 


190 


180 


170 


150 


140 


130 


Vx 


140 


130 


120 


X 


120 


110 


110 


100 

_90_ 
80 


^ 


100 
90 


80 


70 


70 


60 


60 


50 


c^400 


50 


40 


40 


80 


30 


20 


ijOO 


10 


10 


00 


01 


02 

"V  allies 


03 


04 


of  ^  and 


05 


PLATE  VIII. — Coefficients  of  Resistance  of  T-beams. 


DIAGRAMS 


PLATE  IX. — Coefficients  of  Resistance  of  T-beams. 


284 


FORMULAS,  DIAGRAMS,    AND    TABLES.  [CH.  VI. 


fs=i6,ooo 


0.0 


1.00 


0,1 


J 


03 


04 


0.5 


1.00 


.98 


.98 


.94 


sfer 


201 


.94 


.92 


.92 
.90 


.88 


.84 


.82 


.82 


.80 


.80 


150 


190 


180 


170 


160 


150 


140 


140 


130 


130 


120 
110 


110 


100 


•li 


70 


,** 


/t? 


400 


70 


300 


20 


20 


10 


10 


0.1 


Values  of   -T 


and 


0,5 


PLATE  X. — Coefficients  of  Resistance  of  T-beams. 


§  152.] 


DIAGRAMS. 


285 


=  18,000 


PLATE  XI. — Coefficients  of  Resistance  of  T-beams. 


286 


FORMULAS,  DIAGRAMS,   AND  TABLES. 


[Cn.  VI. 


Percentage  of  compressive  steel 
PLATE  XII. — Compressive  Reinforcement  of  Beams. 


§  152.] 


DIAGRAMS. 


2S7 


.02       .04       .06       .08      .,10       .12       .14       .16       .1.8       .20       .22 


.19 
.18 
.17 
.16 
45 
44 
.13 
.12 

•1- 

s.io 

1.09 

1 
.08 

.07 
.06 
.05 
.04 
.03 
.03 
.01 
0 

19 
IS 
.17 
.16 
.15 
.14 
.13 
.13 
.11 
.10 
.09 
.08 
.07 
.Oti 
.05 
.01 
.0:5 
.03 
.01 
0 

& 

M 

h 

^ 

x 

X 

/  e 

X 

x^ 

M 

h 

o 

• 

—  J- 

f  t 

n 

x 

^^ 

. 

^ 

X 

All  Vi 

or 

ilues  of 

M  :  bh-fc 

andd1'./! 

a.re  based 

x 

x 

^ 

»t—  15 

X 

^ 

/ 

X' 

•^ 

/ 

r 

x 

/ 

x 

^x 

'^ 

/ 

x 

X 

x- 

? 

£ 

/ 

/ 

x 

x 

x 

^ 

-^ 

A 

r 

^ 

/ 

x 

/*' 

X 

/ 

V 

^ 

x 

x 

x 

>• 

^ 

t 

^V 

A 

x 

/ 

^ 

x 

X 

x 

-x. 

'^ 

^ 

/ 

/ 

y, 

x 

x 

^ 

x 

^ 

^< 

***'^" 

/ 

/ 

'  / 

> 

/ 

^x 

^ 

•^ 

/ 

/ 

t  / 

<@l/ 

_x 

^ 

x- 

? 

.*•—• 

-" 

/ 

*  / 

/  / 

/ 

/ 

iSJLA 

x 

^ 

X 

^ 

•*"• 

' 

/ 

/  / 

/  , 

/  f 

/ 

Stf 

x 

^^ 

"^ 

/ 

/. 

/  / 

// 

/ 

x 

<si^- 

s 

**' 

> 

'  / 

7, 

/  / 

/  , 

/ 

x 

o^lX 

/, 

/  / 

/  , 

/ 

/ 

, 

/ 

Z 

z 

// 

/  / 

X 

7y 

'// 

z 

'  '  / 

* 

/ 

/// 

// 

// 

2 

1 

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/// 

// 

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I/ 

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\ 

.04        .06       .08       .10       .12       .1.4       .16       .18       .20       .23.      .24 
Values  of  Eccentricity, 


PLATE  XIII. — Flexure  and  Direct  Stress. 


288 


FORMULAS,  DIAGRAMS,   AND  TABLES.  [Cn.  VI. 


i.o 


.6  .8  1.0  1.2  1.4 

Values  of  Eccentricity,  e-t-h 

PLATE  XIV. — Flexure  and  Direct  Stress. 


1.6          1.8 


§  152.] 


DIAGRAMS. 


289 


Percentage  Reinforcement 


1.70 
A.60 

/ 

1.70 
1.60 
1.50 

1.40 
1.30 
1.20 
1.10 
1,0 

j 

. 

/ 

/* 

/ 

/ 

y* 

^/ 

^ 

/ 

/ 

/ 

/ 

/ 

1.40 
/1.30 

i.ao 

/ 

/ 

p 

V 

x 

\/ 

x 

^ 

/ 

/* 

/ 

*4 

^ 

x 

^ 

/ 

/ 

x 

C 

/ 

/ 

x 

1 

/ 

/ 

t< 

^ 

' 

^ 

/ 

/ 

«, 

x 

,xJ 

^ 

r 

/ 

^, 

/ 

x 

/ 

x 

^ 

a^, 

x 

/ 

x 

\ 

x 

x 

/ 

J 

/ 

/ 

x 

> 

^ 

^ 

/ 

* 

/ 

x 

^x1 

^ 

^ 

x* 

/ 

/ 

y 

/ 

•^ 

x 

x 

X* 

/ 

, 

x 

x 

X 

^x 

X 

J 

/ 

/ 

> 

/ 

J 

X 

x 

X 

/ 

/ 

x 

x 

x^ 

•^ 

/ 

/ 

, 

\s 

x 

X 

x^ 

/ 

/ 

/ 

r~ 

s 

' 

^ 

X 

/ 

x 

x 

s 

X 

^ 

** 

^. 

2 

s 

/ 

x 

^> 

^ 

y/ 

x 

^ 

*" 

i 

/x 

> 

^ 

0.0  Ok5  IjQ  1,5  2.0  2S5  3.0 

Percentage  Reinfarcement 


PLATE  XV. — Working  Stresses  in  Columns, 


290 


FORMULAS,  DIAGRAMS,   AND  TABLES.  CH.  VI. 


''-7P' 

M1 
M 

g 
Q 

I   I   I    i   i    I 

iv-Qf«nf 

15 

i  i  i  I 

jl 

w/$ffi$%wffi/$$$ 

^^^^^^/^^^ 

L  ia  bending  moment,  per  unit  width  oi 
section,  causing  circumferential  fiber- 
stress  at  any  distance  r. 
,  is  bending  moment,  per  unit  -wudtih  of 
section,  causing  radial  fiber-stress  at 
any  distance  r. 
is  top  load  per  unit  area, 
is  ordinate  to  proper  dotted  cur.ve. 
l  is  ordinate  to  proper  solid  cur.ve. 
If  q  is  expressed  in  Ibs.  per  sq.  .f  k 
and  r0  in  ft.,  then  M,and  M2will  be  in 
ft.-lbs.  per  ft. 

6I       \ 

\ 

\ 

\ 

\ 

\ 

fr 

\ 

\ 

\ 

\ 

i 

\ 

5!  •  \ 

\ 

\ 

\ 

1  \ 

\ 

\ 

\ 

V— 

V 

i  \  / 

\ 

\  • 

x^ 

\  / 

\ 

\ 

^ 

vv 

10 

A 

\^ 

\ 

'  x- 

•x 

4 

<  \ 

\ 

^^ 

v^ 

\l       f 

\ 

V- 

v-« 

%. 

\/ 

\ 

^^ 

^ 

. 

^ 

'A 

\ 

\ 

\ 

"^^ 

>«^ 

-^^ 

'  \ 

,  —  -i 

^~^ 

\ 

\ 

"^^•^.^ 

^ 

r~+">[ 

3 

j  / 

V 

\ 

_5 

.^.^ 

\ 

""--. 

~.  

B 

~' 

\^- 

-X 

N 

. 

NS^ 

"°N 

;--._ 

5 

i/'\ 

s^J"^ 

~^XL—  "-^ 

x^^ 

4  ^^ 

.  "• 

~"^ 

^^ 

^^^ 

-^-^..^ 

^j 

^^^ 

--  

a 

PLATE  XVI. — Stresses  in  Circular  Slabs. 


345 
Values  of  r  -i-  r^ 


152.] 


DIAGRAMS. 


291 


ijinln. 


r 


.Mjand  M0are  explained  on  Plate  X, 
.  P    is  load  at  periphery  per  unit  length. 
F>    is  ordinate  to  proper  dotted  curve. 
(3    is  ordinate  to  proper  solid  curve. 

If  p  is  expressed  111  Ibs.  per  ft. 

and  r0  in  ft.,  then  iv^and  M2will  be  in 

ft.-lbs.  per  ft. 


PLATE  XVII. — Stresses  in  Circular  Slabs. 


292  FORMULAS,   DIAGRAMS,  AND  TABLES.  [Cn.  VL 

Ii 


§§ 


2    OQ 

£    S 

O 


tfl 


b    I 


00 


m 


£22 


rHt>C005t>-COCOOOOC 
T-  ii—  i(N(MCO'*^OCOI>O:i 


O  T-H 

,-1  1-H  i-H  rH  <N  <N 


1-H  T-l  T-H  1-H  <N  (N  CO 


t>  GO  Tp  O 
ci  (N  COCO 


(NfMiNC^iMCOCOCO^1^1 


5*1 


§152] 


TABLES. 


293 


i 


!l 

00  § 

§  § 

1 1 

2 
S 


M 


CSj  t>»  cOOOOt>t"-l>'O5<MiOOCN£>r-O 

TH  1-1.  W  So  eg  •*  »O  «0t>  C&  O  <N  *o  oo  <M  i> 

'  ^  ^  ri  rH  <N  IN 


^i  ^_i  ^H  i—  i  (N  (N  CO 


"M-?   § 

IS! 

t^^ 


•Ml 

PI  ^^S 


V   S«Q 

C  £3  e5 

-i^i 


id 

£  .2^3 
Q     ^ 


^  ,_  rH  rH  (N 


294 


FORMULAS,  DIAGRAMS,    AND    TABLES. 


VI. 


TABLE  No.  20. 
MATERIALS  REQUIRED  FOR  ONE  CUBIC  YARD  OF  CONCRETE. 


Proportion  of  Mixture. 

Required  for  One  Cubic  Yard. 

Cement. 

Sand. 

Stone. 

Ratio: 
Mortar 

Cement, 
Barrels. 

Sand, 
Cubic  Yards. 

Stone, 
Cubic  Yards. 

Stone 

1 

1 

2.0 

.70 

2.57 

0.39 

0.78 

1 

1 

2.5 

.56 

2.29 

0.35 

0.87 

1 

1 

3.0 

.47 

2.06 

0.31 

0.94 

1 

1.5 

t  2.5 

.71 

2.05 

0.47 

0.78 

1 

1.5 

3.0 

.60 

1.85 

0.42 

0.84 

1 

1.5 

3.5 

.51 

1.72 

0.39 

0.91 

1 

1.5 

4.0 

.44 

1.57 

0.36 

0.96 

1 

2.0 

3.0 

.72 

1.70 

0.52 

0.77 

1 

2.0 

3.5 

.62 

1.57 

0.48 

0.83 

1 

2.0 

4.0 

.54 

1.46 

0.44 

0.89 

1 

2.0 

4.5 

.48 

1.36 

0.42 

0.93 

1 

2.5 

4.0 

.64 

1.35 

0.52 

0.82 

1 

2.5 

4.5 

.57 

1.27 

0.48 

0.87 

1 

2.5 

5.0 

.51 

1.19 

0.46 

0.91 

1 

2.5 

5.5 

.46 

1.13 

0.43 

0.94 

1 

3 

4.5 

.66 

1.18 

0.54 

0.81 

1 

3 

5.0 

.60 

1.11 

0.51 

0.85 

1 

3 

5.5 

.54 

1.06 

0.48 

0.89 

1 

3 

6.0 

.50 

1.00 

0.46 

0.92 

1 

3 

6.5 

,.46 

.96 

0.44 

0.95 

§  152.] 


TABLES. 


295 


TABLE  No.  21.— STRENGTH  OF  FLOOR-SLABS. 
Bold-faced  type,  M  =  %wl2;  light-faced  type,  M 
/c  =  500        /s  =  14,000  £  =  77         p 


' 

-i    ^ 

K 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 
of  Floor  in  Pounds. 

•j 

a 
o      ..; 

"o  oJ2 

•S«J 

a 

|| 

°&J 

Jy 

J     '• 

|| 

r 

Thickness 
crete  be 
Steel,  Ii 

Required 
Steel  p< 
of  Slab, 

Moment  o 
anoe  pe 
of  Slab, 

IJa 

50 

75 

100 

150 

200 

250 

300 

400 

500 

2 

1 

.094 

1400 

24.2 

3.6 

4.4 

3.1 

3.8 

2.8 
3  4 

24 

2.9 

21 

2.6 

1.9 

2.3 

1.7    1.5 

2.1!   1.8 

14 

1.7 

24 

! 

.131 

2800 

30.3 

4.8 
5.9 

4.2 

5.1 

3^8 
4.6 

32   29 

3.9    3.6 

2.6 

3.2 

2.4 

2.9 

21 

2.6 

1.9 

2.3 

3 

1 

.168 

4700 

36.4 

6.0 

7.4 

53 

6.5 

4.8 
5.9 

41 

5.0 

36 

4.4 

33 

4.0 

30 

3.7 

27 

3.3 

24 

2.9 

3* 

i 

.206 

7000 

42.5 

7.1 

8.7 

6.3 

7.7 

57 

7.0 

4.9   4.4 

6.0    5.4 

40 

4.9 

37 

4.5 

33 

4.0 

2.9 

3.6 

4 

i 

.224 

8300 

48.5 

75 

9.2 

6.7 

8.2 

6.1 

7.5 

53 

6.5 

47 

5.8 

43 

5.3 

4.0 

4.9 

35 

4.3 

32 

3.9 

5 

i 

.299 

14800 

60.7 

95 

11.6 

8.5 
10.4 

7.9 

9.7 

6.8 
8.3 

6.2 

7.6 

57 

7.0 

5.2   46 

6.41  5.6 

42 

5.1 

6 

u 

.355 

20900 

72.9 

10  5 

12.8 

9.5 

11.6 

8.8 
10.8 

7.8 
9.6 

7.0 

8.7 

6.5 

8.0 

6.0 

7.5 

54 

6.6 

4.8 
5.9 

7 

H 

.430 

30600 

85.1 

12  2 

14.9 

11.1 

13.6 

10  3 

12.6 

9.3 

11.4 

83 

10.2 

7.7 
9.4 

72 

8.8 

65 

8.0 

5.8 
7.1 

-« 

00  »C5 

u 

14 

.505 
.561 

42100 
52000 

97.4 
IO3.4S 
109.5 

13  7 

16.8 
14  8 

18.1 

12  6 

15.4 
13  7 

16.8 

11.8 

14.4 
12.9 

15.8 

10.  6|  9.6 

12.911.7 
11.510  6 

14.1  12.9 

9.0 

11.0 

9.8 
12.0 

8.3 
10.2 
9.2 
11.2 

7.5 

9.2 
83 

10.2 

6.8 
8.3 
7.6 
9.3 

10 

14 

.636 

66800 

121.7 

16  1 

19.7 

15.0 

18.4 

14.2 

17.4 

12  811  8 

15.614.4 

11.010  3 

13.512.6 

9.3 

11.4 

8.5 
10.4 

12 

14 

.785 

102000 

145.9 

18  9 

23.1 

17.5 

21.4 

16  6115.114.1 

20.3118.  517.  2 

13  1 

16.0 

12  4 

15.2 

11  2 

13.7 

10  3 

12.6 

15,000 


.0056 


2 

1 

.083 

1400 

24.2 

35 

4.3 

31 

3.8 

27 

3.3 

23 

2.8 

20 

2.4 

1.8 
2.2 

1.7 

2.1 

15 

1.8 

1.3 

1.6 

.24 

1 

.117 

2700 

30.8 

4.7 

5.8 

41 

5.0 

37 

4.5 

32 

3.9 

2.8 
3.4 

2.5 

3.1 

23 

2.8 

20 

2.4 

1.8 
2.2 

3 

1 

.150 

4500 

36.4 

5.9 

7.2 

5.2 

6.4 

4.7 

5.8 

4.0 

4.9 

3.6 

4.4 

32 

3.9 

3.0 

3.7 

2.6 

3.2 

24 

2.9 

3* 

1 

.183 

6700 

42.5 

70 

8.6 

6.2 

7.6 

5.6 

6.9 

4.8 
5.9 

43 

5.3 

3.9 

4.8 

36 

4.4 

3.2|  2.9 

3.9    3.6 

4 

1 

.200 

8000 

48.5 

74 

9.0 

6.6 

8.1 

6.0 

7.4 

5.2 

6.4 

46 

5.6 

4.2 

5.1 

3.9 

4.8 

3.4 

4  2 

3.1 

3.8 

5 

1 

.267 

14200 

60.6 

9.2 

11.2 

8.3 

10.2 

7.7 
9.4 

6.7 

8.2 

6.0 

7.4 

5.5 

6.7 

51 

6.2 

i.l 

i.l 

6 

li 

.317 

20100 

72.8 

10  4 

12.7 

9.5 

11.6 

8.8 
10.8 

7.7 
9.4 

7.0 

8.6 

6.4 

7.8 

6.0 

7.4 

53 

6.5 

4.8 
5.9 

7 

H 

383 

29400 

84.9 

12  0 

14.7 

11.1 

13.6 

10.3 

12.6 

9.1 

11.1 

8.3 
10.2 

7.6 

9.3 

7.1 

8.7 

64 

7.8 

5.8 
7.1 

8 

n 

.450 

40500 

97.2 

13.5 

16.5 

12  5 

15.3 

11.7 

14.3 

10  5 

12.8 

9.5 

11.6 

8.8 
10.8 

8.2 
10.0 

74   67 

9.0    8.2 

9 

i* 

500 

50000 

109.3 

14.5 

17.7 

13  5 

16.5 

12  6J11  310  4 

15.4  13.812.7 

9.6!  9.01  8.1 

11.711.0    9.9 

7.4 
9.0 

10 

14 

.567 

64200 

121.5 

15.8 

19.3 

14  7 

18.0 

13  912  611.5 

17.015.4114.1 

10  710  1 

13.1  12.4 

9.1 

11.1 

83 

10.2 

12 

14 

.700 

98000 

145.7 

18  2 

22.2 

17.2 

21.0 

16  314  813  7 

19.9118.  1J16.7 

12  812  110  910  0 
15.7!14.813.3|l2.2 

296 


FORMULAS,  DIAGRAMS,  AND  TABLES. 


CH.  VI 


TABLE  No.  21  (Continued}.— STRENGTH  OF  FLOOR-SLABS. 

Bold-faced  type,  M  =  %wl2 ;  light-faced  type,  M 
3.  /c  =  500        fs=*  16,000  72  =  71         j 


i 

<« 

j,     . 

• 

<T    8 

o 

°  "ShS 

.2  +->,o 

a 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

S*S 

o     <S 

c3  C     . 

80*. 

•^  "^ 

of  Floor  in  Pounds. 

C  fl 

••M 

°~  c 

<J  s-32 

«*-  I-*"1 

Sg 

"•*•§ 

Si.S'-1 

"S  ®"§ 

°  a-§" 

*S 

—  53 
0"° 

e.S'af 

•§12 

111 
»N 

|l? 

•s&J 

'SJCi-! 

50 

75 

100 

150 

200 

250 

300 

400 

500 

H 

|OC 

PH 

a 

^ 

2 

| 

.075 

1300 

24.2 

35 

4.3 

30 

3.7 

2.7 
3.3 

2,3 

2.8 

2.0 

2.4 

1.8 
2.2 

1.7 

2.1 

1.4 

1.7 

1.3 

1.6 

2* 

a 

4 

.105 

2600 

30.3 

4.7 

5.8 

41 

5.0 

37 

4.5 

31 

3.8 

2.8 
3.4 

2.5 

3.1 

23 

2.8 

20 

2.4 

1.8 
2.2 

3 

i 

.135 

4300 

36.4 

5.8 
7.1 

51 

6.2 

46 

5.6 

4.0 

5.0 

3.5 

4.3 

32 

3.9 

2.9 

3.6 

2.6 

3.2 

23 

2.8 

3* 

i 

.165 

6500 

42.5 

6.8 
8.3 

61 

7.5 

5.5 

6.7 

4.8 
5.9 

4.2 

5.1 

3.8 
4.6 

3.6 

4.4 

3.1 

3.8 

2.8 
3.4 

4 

i 

.180 

7700 

48.5 

7.2 

8.8 

64 

7.8 

5.9 

7.2 

51 

6.2 

4.5 

5.5 

4.1 

5.1 

3.8 
4.6 

34 

4.2 

31 

3.8 

5 

i 

.239 

13700 

60.7 

9.1 

11.1 

8.2 
10.0 

75 

9.2 

6.6 

8.1 

5.9 

7.2 

5.4 

6.6 

5.0 

6.1 

44 

5.4 

4.0 

4.9 

6 

ii 

.284 

19300 

72.7 

10  3 

12.6 

9.4 

11.5 

8.7 
10.6 

7.6 

9.3 

6.9 

8.4 

6.3 

7.7 

5.9 

7.2 

5.2 

6.4 

4.7 

5.8 

7 

ii 

.344 

28300 

84.8 

11.9 

14.5 

10.9 

13.3 

10.1 

12.4 

9.0 

11.0 

8.2 
10.0 

7.5 

9.2 

7.0 

8.6 

6.3 

7.7 

5.7 
7.0 

8 

U 

.404 

39000 

97.0 

13  4 

16.4 

12  3 

15.0 

11.5 

14.1 

10  3 

12.6 

9.4 

11.5 

8.7 
10.6 

8.1 
9.9 

73 

8.9 

6.6 

8.1 

9 

i* 

.449 

48100 

109.1 

14.3 

17.5 

13  3 

16.3 

12.4 

15.2 

11.2 

13.7 

10.2 

12.5 

9.5 

11.6 

8.9 
10.9 

8.0 
9.8 

7.3 

8.9 

10 

H 

.509 

61800 

121.3 

15.6 

19.1 

13.6 

17.9 

13.7 

16.7 

12.4 

15.2 

11.4 

13.9 

10.6 

12.9 

9.9 

12.1 

8.9 
10.9 

8.2 
10.0 

12 

ii 

.629 

94400 

145.5 

17.9 

21.8 

16.9 

20.7 

16.0 

19.5 

14  6 

17.9 

13.5 

16.5 

12  6 

15.4 

11.9 

14.5 

10  7 

13.1 

9.9 

12.1 

c  =  500 


18,000 


=  66 


2 

1 

061 

1200 

24.2 

33 

4.0 

2.9 

3.6 

26 

3.2 

2.2 

27 

19 

2.3 

17 

2.1 

16 

2.0 

1.4 

1  .7 

1.3 

1.6 

2i 

1 

.086 

2400 

30.3 

45 

5.5 

39 

4.8 

3.5 

4.3 

30 

3.7 

26 

3.2 

24 

2.9 

2.2 

2~7 

1.9 

2.3 

1.7 

2.1 

3 

! 

.110 

4000 

36.4 

56 

6.9 

49 

6.0 

4.4 

5.4 

3.8 
4.6 

3.4 

4.2 

30 

3.7 

2.8 
3.4 

25 

3.1 

2.2 

2.7 

3i 

1 

135 

6000 

42.4 

6.6 

8.1 

5.9 

7.2 

53 

6.5 

46 

5.6 

4.0 

4.9 

37 

4.5 

34 

4.2 

3.0 

3.7 

27 

3.3 

4 

l 

.147 

7200 

48.4 

70 

8.6 

6.2 

7.6 

57 

7.0 

4.9 

6.0 

4.4 

5.4 

40 

4.9 

37 

4.5 

3.3 

4.0 

2.9 

3.6 

5 

1 

196 

12700 

60.5 

8.8 
10.8 

7.9 

9.7 

7.3 

8.9 

6.4 

7.8 

5.7 

7.0 

52 

6.4 

4.8 

5-2 

4.3 

5.3 

3.9 

4.8 

6 

H 

233 

18000 

72.6 

9.8 
12.0 

9.0 

11.0 

8.3 
10.2 

73 

8.9 

6.6 

8.1 

6.1 

7.5 

5  .  7 

7.0 

5.0 

6.1 

4.6 

5.6 

7 

ii 

.282 

26300 

84.7 

11  4 

13.9 

10  4 

12.7 

9.7 

11.9 

8.6 
10.5 

7.8 
9.6 

7.2 

8.8 

6.7 

8.2 

6.0 

7.4 

5.5 

6.7 

8 

n 

331 

36300 

96.8 

12.8 

15.6 

11.8 

14.4 

11.0 

13.5 

9.9 

12.1 

9.0 

11.0 

8.3 
10.2 

7.8 
9.6 

7.0 

8.6 

6.4 

7.8 

9 

H 

368 

44800 

108  9 

13  6 

16.6 

12  7 

15.5 

11.9 

14.5 

10.7 

13.1 

9.8 
12.0 

9.1 

11.1 

8.5 
10.4 

77 

9.4 

7.0 

8.6 

10 

i* 

.417 

57500 

121.1 

14.9 

18.2 

13.9 

17.0 

13.1 

16.0 

11.9 

14.5 

10  9 

13.3 

10.1 

12.4 

9.5 

11.6 

8.6 
10.5 

7.9 

9.7 

12 

i* 

.515 

87700 

145.3 

17  1 

20.9 

16  2 

19.8 

15  414  0 

18.817.1 

13.0 

15.9 

12  1 

14.8 

11  4 

13.9 

10  4 

12.7 

9.5 

11.6 

§  152.] 


TABLES. 


297 


TABLE  No.  21  (Continued)— STRENGTH  OF  FLOOR-SLABS. 

Bold-faced  type,  M  =  %wl2\  light-faced  type,  M 
5.  /c  =  600        fs  =  14,000  #=102        y 


i 

i 

"o^c 

i^j 

&  . 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

1-g     OS 

^  o 

8  o  « 

^  "ft 

of  Floor  in  Pounds. 

c  c     <*-  *  — 

<*t^ 

"*        HH 

5s£ 

C*S 

—  53 

0*° 

|f 

o/__d 
|lo 

fi-s 

"o  g> 

50 

75 

100 

150 

200 

250 

300 

400 

500 

H 

r 

K 

5 

r 

2 

a 

4 

126 

1900 

24.3 

41 

5.0 

3.6 

4.4 

32 

3.9 

27 

3.3 

24 

2.9 

22 

2.7 

20 

2.4 

17 

2.1 

1.6 

2.0 

*l 

| 

.176 

3800 

30.4 

5.6 

6.9 

4.9 

6.0 

4.4 

5.4 

3.7 

4.5 

3.3 

4.0 

30 

3.7 

27 

3.3 

24 

2.9 

22 

2.7 

3 

I 

.226 

6200 

36.5 

6.9 

8.4 

6.1 

7.5 

5.5 

6.7 

47 

5.8 

42 

5.1 

3.8 
4.6 

35 

4.3 

31 

3.8 

2.8 
3.4 

** 

1 

.277 

9300 

42.7 

8.2 
10.0 

72 

8.8 

6.6 

8.1 

57 

7.0 

5.0 

6.1 

4.6 

5.6 

42 

5.1 

3.7 

4.5 

34 

4.2 

4 

1 

.302 

11000 

48.7 

8.6 
10.5 

7.7 
9.4 

70 

8.6 

6.1 

7.5 

54 

6.6 

50 

6.1 

4.6 

5.6 

40 

4.9 

37 

4.5 

5 

1 

.403 

19600 

60.9 

10  8 

13.2 

9.8 
12.0 

9.0 

11.0 

7.9 

9.7 

7.1 

8.7 

65 

8.0 

6.0 

7.4 

53 

6.5 

4.8 
5.9 

6 

H 

.478 

27700 

73.1 

12.2 

14.9 

11  2 

13.7 

10  3 

12.6 

9.1 

11.1 

8.2 
10.0 

75 

9.2 

70 

8.6 

6.2 

7.6 

57 

7.0 

7 

H 

.579 

40600 

85.4 

14  1 

17.2 

13  0 

15.9 

12  1 

14.8 

10.7 

13.1 

9.7 

11.9 

9.0 

11.0 

8.4 
10.3 

7.5 

9.2 

6.8 
8.3 

8 

H 

.680 

55900 

97.8 

15  9 

19.4 

14  7 

18.0 

13.7 

16.8 

12.2 

14.9 

11.2 

13.7 

10  3 

12.6 

.9.7 

11.9 

8.6 
10.5 

7.9 

9.7 

9 

« 

.755 

69000 

109.8 

16.9 

20.6 

15  7 

19.2 

14.8 

18.1 

13.3 

16.2 

12  2 

14.9 

11  3 

13.8 

10.6 

12.9 

9.5 

11.6 

8.7 
10.6 

10 

H 

.856 

88600 

122.0 

18  5 

22.6 

17.3 

21.1 

16.3 

19.9 

14.7 

18.0 

13  5 

16.5 

12.6 

15.4 

11.8 

14.4 

10.6 

12.9 

9.7 

11.9 

12 

i* 

1  057 

135300 

146.4 

21.4 

26.2 

20.1 

24.6 

19.1 

23.3 

17.4 

21.3 

16.1 

19.7 

15.0 

18.3 

14.2 

17.3 

12.8 

15.7 

11.8 

14.4 

fs  =  15,000 


o 

i 

112 

1800 

24.2 

41 

5.0 

35 

4.3 

32 

3.9 

2.7 

3.3 

23 

2.8 

21 

2.6 

1.9 

2.3 

17 

2.1 

1.5 

1.8 

2i 

1 

.157 

3600 

30.4 

55 

6.7 

4.8 
5.9 

4.3 

5.3 

37 

4.5 

32 

3.9 

29 

3.6 

27 

3.3 

24 

2.9 

21 

2.6 

3 

1 

.202 

6000 

36.5 

6.8 
8.3 

6.0 

7.4 

54 

6.6 

4.6 

5.6 

41 

5.0 

3.7 

4.5 

3.4 

4.2 

30 

3.7 

27 

3.3 

3^ 

! 

.247 

8900 

42.7 

8.0 
9.8 

71 

8.7 

6.5 

8.0 

5.6 

6.9 

50 

6.1 

4.5 

5.5 

42 

5.1 

37 

4.5 

33 

4.0 

4 

1 

.270 

10600 

48.7 

8.5 
10.4 

7.6 

9.3 

6.9 

8.4 

6.0 

7.4 

53 

6.5 

4.9 

6.0 

45 

5.5 

40 

4.9 

3.6 

4.4 

5 

1 

360 

18900 

60.9 

10  7 

13.1 

9.6 

11.7 

8.8 
10.8 

7.7 
9.4 

7.0   6.4 
8.6|  7.8 

5.9 

7.2 

52 

6.4 

47 

5.8 

6 

H 

.427 

26700 

73.1 

12.0 

14.7 

11.0 

13.5 

10.1 

12.4 

8.9 
10.9 

8.1 
9.9 

74 

9.0 

6.9 

8.4 

6.1 

7.5 

5.6 

6.9 

7 

U 

.517 

39100 

85.5 

13.9 

17.0 

12.8 

15.7 

11.9 

14.5 

10.5 

12.8 

9.6 

11.7 

8.8 
10.8 

8.2 
10.0 

73 

8.9 

6.7 

8.2 

8 

n 

.607 

53800 

97.9 

15.6 

19.1 

14.4 

17.6 

13.5 

16.5 

12.1 

14.8 

11.010  2 

13.512.5 

9.5 

11.6 

8.5 
10.4 

7.8 
9.6 

9 

H 

.675 

66400 

109.6 

16.715.5 

20.4il9.0 

14.613.1 

17.816.0 

12.011.1 

14.7  13.6 

10.4 

12.7 

9.4 

11.5 

8.5 
10.4 

10 

i* 

.765 

85300 

121.8 

18.217.0 

22.220.8 

16.0 

19.5 

14.5 

17.7 

13  3 

16.2 

12  4 

15.2 

11.6 

14.2 

10.4 

12.7 

9.6 

11.7 

12 

H 

.945 

130200 

146.2 

21.119.8 

25.824.2 

18  817.1 

23.0120.9 

15.814.8 

19.318.1 

14.0 

17.1 

12  6 

15.4 

11.6 

14.2 

298 


FORMULAS,  DIAGRAMS,  AND    TABLES. 


CH.  VI. 


TABLE  No.  21   (Continued)— STRENGTH  OF  FLOOR-SLABS. 
Bold-faced  type,  M  =  %wl2;  light-faced  type,  M 
/c  =  600        /s=  16,000  R  =  95        p 


gg 

§ 

•g^d 

<M 

1 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

1"! 

Jr"    J 

f^  ELI  O* 

dfcl 

If 

of  Floor  in  Pounds. 

"oM 

.    g 

<J  t-^ 

*-.  s-^"1 

5» 

11 

11 

111 

ggl 
1S<S 

i!s 

50 

75 

100 

150 

200 

250 

300 

400 

500 

H 

H 

(2 

*• 

^ 

2 

1 

.101 

1800 

24.2 

4.0 

4.9 

3.5 

4.3 

31 

3.8 

2.6 

3.2 

23 

2.8 

21 

2.6 

1.9 

2.3 

1.7 

2.1 

1.5 

1.8 

2* 

! 

.142 

3500 

30.3 

54 

6.6 

47 

5.8 

42 

5.1 

36 

4.4 

32 

3.9 

2.9 

3.6 

27 

3.3 

23 

2.8 

21 

2.6 

3 

! 

.182 

5800 

36.4 

6.7 

8.2 

59 

7.2 

53 

6.5 

4.5 

5.5 

40 

4.9 

37 

4.5 

34 

4.2 

3.0 

3.7 

27 

3.3 

3i 

1 

.223 

8600 

42.6 

7.9 

9.7 

70 

8.6 

63 

7.7 

55 

6.7 

49 

6.0 

44 

5.4 

41 

5.0 

3.6 

4.4 

33 

4.0 

4 

1 

.243 

10300 

48.6 

8.3 
10.2 

7.4 

9.0 

6.8   5.9 

8.3    7.2 

52 

6.4 

4.8 
5.9 

4.4 

5.4 

39 

4.8 

35 

4.3 

5 

1 

324 

18300 

60.8 

10.5 

12.8 

9.5 

11.6 

8.7 
10.6 

7.6   6.8 
9.3    8.3 

63 

7.7 

5.8 
7.1 

51 

6.2 

47 

5.8 

6 

H 

.385 

25700 

73.0 

11.8 

14.4 

10.7 

13.1 

99 

12.1 

8.7    7.9 

10.6   9.7 

73 

8.9 

6.8 
8.3 

6.0 

7:4 

55 

6.7 

7 

U 

466 

37700 

85.2 

13  6 

16.6 

12.5 

15.3 

11  6 

14.2 

10  3   9.4 

12.611.5 

8.7 
10.6 

8.1 
9.9 

7.2 

8.8 

65 

8.0 

8 

H 

.547 

52000 

97.7 

15.3 

18.7 

14.2 

17.3 

13  2 

16.1 

11.810.810.0 

14.413.212.2 

9.4 

11.5 

8.3 

10.2 

7.6 

9.3 

9 

l* 

.608 

64200 

109.4 

16.4 

20.0 

15.2 

18.6 

14  3 

17.5 

12.811.810.910.2 

15.714.413.312.5 

92 

11.2 

8.4 
10.3 

10 

li 

.689 

82400 

121.6 

17.9 

21.9 

16.7 

20.4 

15  7 
19.2 

14  213.112.2 

17.316.014.9 

11  4 

13.9 

10.2 

12.5 

9.4 

11.5 

12 

U 

851 

125800 

146.2 

20.6 

25.2 

19.4 

23.7 

18  4 

22.5 

16  815.514.513.7 

20.519.017.7116.8 

12.4 

15.2 

11.4 

13.9 

/c  =  600 


18,000 


2 

! 

.083 

1700 

24  1 

3.9 

4.8 

3.3 

4.0 

30 

3.7 

25 

3.1 

22 

2.7 

20 

2.4 

1.8 
2.2 

1.6 

2.0 

1.4 

1.7 

2i 

! 

117 

3300 

30.3 

52 

6.4 

4.5 

5.5 

41 

5.0 

35 

4.3 

3.1 

3.8 

2.8 
3.4 

2.6 

3.2 

22 

2.7 

20 

2.4 

3 

1 

150 

5400 

36.4 

65 

8.0 

57 

7.0 

5.1 

6.2 

44 

5.4 

39 

4.8 

35 

4.3 

33 

4.0 

29 

3.6 

2.6 

3.2 

3i 

1 

.183 

8100 

42.6 

7.6 

9.3 

6.8 
8.3 

6.1 

7.5 

5.3 

6.5 

47 

5.8 

4.3 

5.3 

4.0 

4.9 

35 

4.3 

31 

3.8 

4 

1 

.200 

9600 

48.6 

8.1 
9.9 

72 

8.8 

6.6 

8.1 

5.7 

7.0 

5.1 

6.2 

4.6 

5.6 

4.3 

5.3 

3.8 
4.6 

34 

4.2 

5 

1 

.267 

17100 

60.7 

10.1 

12.4 

9.2 

11.2 

8.4 
10.3 

74 

9.0 

6.6 

8.1 

6.1 

7.5 

5.6 

6.9 

5.0 

6.1 

45 

5.5 

6 

H 

.317 

24100 

72.8 

11  4 

13.9 

10.4 

12.7 

9.6 

11.7 

8.5 
10.4 

7.7 
9.4 

71 

8.7 

6.6 

8.1 

5.8 
7.1 

5.3 

6.5 

7 

H 

383 

35300 

85.2 

13.2 

16.1 

12  1 

14.8 

11.3 

13.  S 

10.0 

12  2 

9.1 

11.1 

8.4 
10.3 

7.8 
9.6 

7.0:  6.3 

8.6  7.7 

8 

U 

450 

48600 

97.5 

14.8 

18.1 

13  7 

16.8 

12.8 
15.7 

11  4 

13.9 

10  4 

12.7 

9.6 

11.7 

9.0 

11.0 

8.1  74 
9.9  9.0 

9 

.1* 

.500 

60000 

109  2 

15.9 

19.4 

14.7 

18.0 

13  8 

16.9 

12  4 

15.2 

11.4 

13.9 

10.6!  9.9 

12.912.1 

8.9  8.1 
10.  9  9.9 

10 

1* 

.567 

77100 

121.3 

17.3 

21.1 

16.2 

19.8 

15  2 

18.6 

13  8 

16.9 

12.6 

15.4 

11.811.0 

14.4  13.5 

9.9  9.1 

12.  I'll.  1 

12 

H 

700 

117600 

145.7 

20  0 

24.5 

18.8 
23.0 

17.8 
21.7 

16  .315.1 

19.9,18.5 

14.113.3 

17.216.2 

12.011  0 

14.713.5 

§  152.] 


TABLES. 


299 


9. 


TABLE  No.  21    (Continued}—  STRENGTH  OF  FLOOR-SLABS. 
Bold-faced  type,  M  =  %wP;  light-faced  type,  M 
/c  =  700        f,  =  14,000  #=129         p 


B 

c 

•83  d 

|   j 

a 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

|1 

o     <S 

o3  o^"*. 

<6  o"^1 

•So" 

of  Floor  in  Pounds. 

*o  o  u 

!*£ 

2^« 

.2  § 

'£•% 

1*  . 

•3  P..Q 

^  ft'0" 

"3  g 

I"® 

1*1" 

fl? 

igl 

O  03*0 

1-3  *     ' 

50 

75 

100 

150 

200 

250 

300 

400 

500 

H 

p 

K 

5 

^ 

2 

i 

.161 

2400 

24.4 

4.7 

5.8 

4.0 

4.9 

3.6 

4.4 

3.0 

3.7 

2.7 

3.3 

2.4 

2.9 

22 

2.7 

1.9 

2.3 

1.7 

2.1 

81 

i 

.225 

4700 

30.6 

6.3 

7.7 

5.5 

6.7 

4.9 

6.0 

4.2 

5.1 

37 

4.5 

34 

4.2 

3.1 

3.8 

2.7 

3.3 

2.4 

2.9 

3 

1 

.289 

7800 

36.7 

7.8 
9.6 

6.8 
8.3 

6.2 

7.6 

5.3 

6.5 

4.7 

5.8 

43 

5.3 

3.9 

4.8 

3.5 

4.3 

3.1 

3.8 

3* 

f 

.354 

11700 

43.0 

9.2 
11.2 

8.1 
9.9 

74 

9.0 

6.4 

7.8 

57 

7.0 

52 

6.4 

4.8 
5.9 

42 

5.1 

3.8 
4.6 

4 

i 

.386 

13900 

48.9 

9.7 

11.9 

8.6 
10.5 

7.9 

9.7 

6.8 
8.3 

6.1 

7.5 

56 

6.9 

51 

6.2 

45 

5.5 

4.1 

5.0 

5 

i 

514 

24700 

61.1 

12  2 

14.9 

11.0 

13.5 

10.1 

12.4 

8.8 
10.8 

7.9 

9.7 

73 

8.9 

6.7 

8.2 

6.0 

7.4 

5.4 

6.6 

6 

H 

.611 

34800 

73.5 

13  7 

16.8 

12  5 

15.3 

11  5 

14.1 

10.2 

12.5 

9.2 

11.2 

8.5 
10.4 

7.9 

9.7 

7.0 

8.6 

6.4 

7.8 

m 

7 

li 

.739 

51000 

85.7 

15.8 

19.3 

14.6 

17.8 

13.5 

16.5 

12.0 

14.7 

10.9 

13.3 

10.1 

12.4 

9.4 

11.5 

8.4 
10.3 

7.6 

9.3 

8 

li 

.868 

70300 

98.1 

17.8 
21.7 

16.5 

20.1 

15.4 

18.8 

13.7 

16.8 

12  5 

15.3 

11.6 

14.2 

10.9 

13.3 

9.7 

11.9 

8.9 
10.9 

9 

li 

.964 

86800 

110.3 

19.0 

23.2 

17.7 

21.6 

16.6 

20.3 

14.9 

18.2 

13.7 

16.8 

12  7 

15.5 

11.9 

14.5 

10.6 

12.9 

9.7 

11.9 

10 

4 

1.093 

111500 

122.6 

20.8 

25.5 

19.4 

23.7 

18  3 

22.3 

16.5 

20.1 

15.2 

18.6 

14.1 

17.2 

13.3 

16.2 

11.9 

14.5 

10.9 

13.3 

12 

u 

1  350 

170100 

147.3 

24  0 

29.3 

22.6 

27.6 

21  5 

26.3 

19  6 

24.0 

18  1 

22.1 

Ifi.9|15.9jl4.  4 

20.6J19.4  17.6 

13.2 

16.1 

15,000 


=  .0096 


2 

1 

.144 

2300 

24.4 

46 

5.6 

4.0 

4.9 

3.5 

4.3 

30 

3.7 

2.6 

3.2 

24 

2.9 

22 

2.7 

1.9 

2.3 

1.7 

2.1 

2* 

1 

.202 

4600 

30.6 

61 

7.5 

5.4 

6.6 

4.8 
5.9 

41 

5.0 

3.6 

4.4 

33 

4.0 

30 

3.7 

2.7 

3.3 

24 

2.9 

3 

! 

.259 

7600 

36.7 

7.6 

9.3 

6.7 

8.2 

6.1 

7.5 

52 

6.4 

46 

5.6 

42 

5.1 

39 

4.8 

3.4 

4.2 

31 

3.8 

31 

i 

.317 

11300 

42.9 

9.0 

11.0 

8.0 
9.8 

73 

8.9 

63 

7.7 

5.6  5.1 

6.91  6.2 

47 

5.8 

4.1 

5.0 

3.7 

4.5 

4 

-j 

.346 

13400 

48.8 

9.5 

11.6 

8.5 
10.4 

7.8 
9.6 

6.7 

8.2 

6.0 

7.4 

56 

6.9 

51 

6.2 

45 

5.5 

4.0 

4.9 

5 

i 

.461 

23900 

61.1 

12.0 

14.7 

10.8 

13.2 

9.9 

12.1 

8.7 
10.6 

7.8 
9.6 

71 

8.7 

6.6 

8.1 

5.9 

7.2 

5.3 

6.5 

6 

H 

.548 

33700 

73.3 

13.5 

16.5 

12.3 

15.0 

11.4 

13.9 

10.0 

12.2 

9.1 

11.1 

8.3 
10.2 

7.7 
9.4 

6.9 

8.4 

63 

7.7 

7 

« 

.663 

49300 

85.5 

15.6 

19.1 

14.3113.4 

17.516.4 

11.8 

14.4 

10.7 

13.1 

9.9 

12.1 

9.2 

11.2 

8.2 
10.0 

7.5 

9.2 

8 

U 

.778 

68000 

97.9 

17.5 

21.4 

16.215.1 

19.818.5 

13.5 

16.5 

12.311.4 

15.013.9 

10  7 

13.1 

9.5 

11.6 

8.7 
10.6 

9 

li 

.865 

83900 

110.1 

18.7 
22.9 

17.416.3 

21.319.9 

14.7 

18.0 

13.4 

16.4 

12.5 

15.3 

11.7 

14'.  c 

10.5 

12.8 

9.6 

11.7 

10 

li 

.980 

107800 

122.3 

20  4 

25.0 

19.1 

23.3 

18.0 

22.0 

16.2 

19.3 

14.9 

18.2 

13.9 

17.0 

13  0 

15.9 

11  7 

14.3 

10  7 

13.1 

12 

ii 

1.210 

164500 

146.9 

23.6 

28.8 

22.221.1 

27.225.8 

19.2 

23.5 

17.8 
21.7 

16.615.7 

20.3)19.2 

14.2 

17.3 

13  0 

15.9 

300 


FORMULAS,  DIAGRAMS,  AND    TABLES.  [On.  VI. 


TABLE  No.  21  (Continued)— STRENGTH  OF  FLOOR-SLABS. 

.     Bold-faced  type,  M  =  %wl2-,  light-faced  type,  M  = 
11.  /c  =  700        /s=16,000  #=120        f 


li 

a 

an 

o^ 

\»t 

t-i 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

oj  o 

fe  s 

o3  O     . 

<u  o  '• 

^O  •+£ 

of  Floor  in  Pounds. 

•gJl 

|^ 

5^~ 

££ 

i 

i 

si's 

M 

1 

^COH 

50 

75 

100 

150 

200 

250 

300 

400 

500 

2 

1 

.130 

2300 

24.3 

45 

5.5 

39 

4.8 

35 

4.3 

2.9 

3.6 

26 

3.2 

23 

2.8 

21 

2.6 

19 

2.3 

1.7 

2.1 

8* 

! 

.182 

4400 

30.5 

6.1 

7.5 

53 

6.5 

4.8 
5.9 

4.0 

4.91 

36 

4.4 

32 

3.9 

30 

3.7 

26 

3.2 

24 

2.9 

3 

1 

.234 

7300 

36.6 

7  .5 

9.2 

6.6 

8.1 

60 

7.4 

51 

6.2 

45 

5.5 

41 

5.0 

3.8 
4.6 

33 

4.0 

3.0 

3.7 

3* 

I 

.286 

10900 

42.9 

8.9 
10.9 

7.9 

9.7 

72 

8.8 

6.2 

7.6 

5.5 

6.7 

5.0 

6.1 

46 

5.6 

4.0 

4.9 

3.7 

4.5 

4 

1 

.312 

13000 

48.8 

93 

11.3 

8.4 
10.3 

7.6 

9.3 

6.6 

8.1 

5.9 

7.2 

54 

6.6 

5.0 

6.1 

44 

5.4 

40 

4.9 

5 

1 

.416 

23100 

61.0 

11.8 

14.4 

10  6 

12.9 

9.8 
12.0 

8.5 
10.4 

7.7 
9.4 

7.0 

8.6 

6.5 

8.0 

5.8 
7.1 

52 

6.4 

6 

H 

.494 

32600 

73.2 

13  3 

16.2 

12.1 

14.8 

11.2 

13.7 

9.9 
12.1 

8.9 
10.9 

8.2 
10.0 

7.6 

9.3 

6.8 
8.3 

6.2 

7.6 

7 

U 

.598 

47700 

85.4 

15  4 

18.8 

14.1 

17.2 

13  1 

16.0 

11.610.6 

14.212.9 

9.8 
12.0 

9.1 

11.1 

8.1 
9.9 

7.4 

9.0 

8 

U 

.702 

65800 

97.7 

17.3 

21.1 

15.9 

19.4 

14  9 

18.2 

13  3 

16.2 

12.1 

14.8 

11.2 

13.7 

10  5 

12.8 

94 

11.5 

8.6 
10.5 

9 

I* 

.780 

81200 

109.9 

18.4 

23.0 

17  1 

20.9 

16.1 

19.7 

14  4 

17.6 

13.2 

16.1 

12.3 

15.0 

11  5 

14.1 

10.3 

12.6 

94 

11.5 

10 

u 

.884 

104300 

122.1 

20  1 

24.6 

18.8 
23.0 

17.7 

21.6 

16  0 

19.5 

14.7 

18.0 

13  7 

16.8 

12.9 

15.8 

11.610  6 

14.2J12.9 

12 

H 

1  092 

159200 

146.6 

23.2 

28.3 

21.9 

26.8 

20.8 
25.5 

18  9 

23.1 

17.5 

21.4 

16  4 

20.0 

15.4 

18.8 

14.0 

17.0 

12.8 

15.7 

12.                /c  =  700        /„  =  18,000                   72  =  113         p  =  .0072 

2 

1 

.107 

2100 

24  3 

4.4 

5.4 

3.8 
4.6 

3.4 

4.2 

2.8 
3.4 

2.5 

3.1 

23 

2.8 

2.1 

2.6 

1.8 
2.2 

1.6 

2.0 

»* 

! 

.150 

4200 

30.4 

59 

7.2 

51 

6.2 

46 

5.6 

39 

4.8 

3.5 

4.3 

31 

3.8 

2.9 

3.6 

2.5 

3.1 

23 

2.8 

3 

1 

.193 

6900 

36.4 

73 

8.9 

6.4 

7.8 

5.8 
7.1 

5.0 

6.1 

4.4 

5.4 

4.0 

4.9 

37 

4.5 

3.2 

3.9 

2.9 

3.6 

3i 

1 

.236 

10300 

42.7 

8.6 
10.5 

7.6 

9.3 

6.9 

8.4 

6.0 

7.4 

53 

6.5 

4.8 
5.9 

45 

5.5 

39 

4.8 

35 

4.3 

4 

1 

.258 

12200 

48.6 

91 

11.1 

8.1 
9.9 

74 

9.0 

64 

7.8 

5.7 

7.0 

52 

6.4 

4.8 
5.9 

4.3 

5.3 

38 

4.6 

5 

1 

.344 

21700 

60.8 

11.4 

13.9 

10  3 

12.6 

9.5 

11.6 

83 

10.2 

7.5 

9.2 

6.8 
8.3 

6.3 

7.7 

56 

6.9 

51 

6.2 

6 

H 

.408 

30600 

72.9 

12.8 

15.7 

11.7 

14.3 

10.9 

13.3 

9.6 

11.7 

8.6 
10.5 

7.9 

9.7 

7.4 

9.0 

6.6 

8.1 

6.0 

7.4 

7 

U 

.494 

44800 

85.4 

14.8 

18.1 

13  6 

16.6 

12.7 

15.5 

11.3 

13.8 

10.2 

12.5 

9.4 

11.5 

8.8 
10.8 

7.8 
9.6 

71 

8.7 

8 

H 

.580 

61800 

97.9 

16.7 

20.4 

15  4 

18.8 

14.5 

17.7 

12.9 

15.8 

11.7 

14.3 

10.9 

13.3 

10.2 

12.5 

9.1 

11.1 

83 

10.2 

9 

1» 

.644 

76300 

109.6 

17.9 

21.9 

16.6 

20.3 

15.6 

19.1 

14.0 

17.1 

12.8 

15.7 

11.9 

14.5 

11.1 

13.6 

10.0 

12.2 

9.1 

11.1 

10 

« 

.73C 

98000 

121.8 

19.5 

23.8 

18.2 
22.2 

17  2115.5 

21.  0)19.0 

14.3 

17.5 

13  3 

16.2 

12  5 

15.3 

11.2 

13.7 

10.3 

12.6 

12 

U 

.902 

149500 

146  3 

22.621.2 

27.625.9 

20.118.417  0 

24.5!22.5|20.8 

15  9 

19.4 

15.0 

18.3 

13.5 

16.5 

12  4 

15.2 

CHAPTER  VII. 
BUILDING  CONSTRUCTION. 

153.  Division   of  the  Subject. — The  various  elements  of 
building   construction    relating   to    reinforced-concrete    design 
may  be  grouped  under  the  following  heads:    (1)  Beams  form- 
ing a  continuous  surface,  as  floor-  and  roof -slabs;    (2)  Floor- 
beams  and  girders;   (3)  Columns;   (4)  Footings;   (5)  Walls  and 
partitions.     In  the  discussion  of  these  various  elements  con- 
sideration will  be  given  to  the  determination  of  stresses,  the 
design  of  the  members,  and  the  arrangement  of  connective 
details. 

154.  General  Arrangement  of  Concrete  Floors. — Two  gen- 
eral types  of  floors  may  be  considered:    (1)  that  in  which  the 
floor- slab  is  supported  on  steel  beams,  and  (2)  that  in  which 
concrete  beams  are  used,  the  floor  of  the  entire  structure  being 
of  a  monolithic  character.     In  the  former  case  the  steel  skele- 
ton consists  of  columns,  girders,  and  cross-beams,  the  beams 
being  spaced    commonly  about  6  feet  apart.    The    floor-slab 
is  supported  mainly  by  the  cross-beams.    The  same  variety 
of  arrangements  is  used  in  the  case  of  all-concrete  structures, 
the  cross-beams  being  spaced  usually  from  four  to  six  feet 
apart.    The  cross-beams  may  in  this  case  be  entirely  omitted, 
giving  span  lengths  of  15  to  20  feet.    Sometimes,  also,  the 
cross-beams  are  inserted  only  at  columns,  forming  a  nearly 
square  panel  of  the  floor-slab,  which  is  then  considered  as  sup- 
ported on  four  sides. 

155.  Stresses  in  Continuous  Beams.— Since  floor-slabs  and 
beams  are  commonly  designed  to  act  as  continuous  beams 
it  is  important  to  investigate  the  possible  stresses  under  such 
conditions,    although   exact   calculation   is   impracticable   and 
unnecessary.     In   the   case   of   floor-slabs   there   is   usually   a 
large  number  of  consecutive  spans  and  the  loading  producing 

301 


302  BUILDING   CONSTRUCTION.  [Cn.  VII. 

the  theoretical  maximum  moments  at  various  points  would 
involve  unreasonable  assumptions  as  to  position  of  live  loads. 
Sufficiently  exact  analysis  may  be  arrived  at  by  considering 
certain  simple  cases. 

Moments  in  Beams  of  Two  and  Three  Spans. — Inasmuch  as 
the  conditions  for  a  theoretical  maximum  are  more  likely  to 
occur  in  beams  of  two  or  three  spans  than  where  the  number  of 
spans  is  large,  an  exact  analysis  will  be  made  of  maximum 
moments  at  all  points  for  the  beam  of  two  spans  and  for  the 
beam  of  three  spans.  The  spans  will  be  assumed  equal  and 
the  beam  considered  as  continuous  but  freely  supported  at 
all  points.  Assume  the  dead  load  to  be  a  uniformly  distributed 
load,  =w  per  lineal  foot,  and  the  live  load  to  be  also  a  uniform 
load,  =p  per  lineal  foot,  but  distributed  over  such  portions 
of  the  beam  as  to  cause  a  maximum  moment  at  the  given 
section.  The  maximum  moment  is  readily  calculated  by 
means  of  the  usual  continuous  girder  formulas.  The  calcula- 
tion will  not  be  repeated  here.  The  results  are  shown  graphi- 
cally to  scale  in  Figs.  71  and  72.  The  dotted  lines  relate  to 
dead-load  effects  and  the  dashed  lines  to  live-load  effects. 
The  span  length  and  the  load  per  foot  are  assumed  equal  to 
unity.  If  /  =  span  length,  the  true  moments  will  be  found  by 
multiplying  the  proper  ordinates  by  wl2  or  by  pi2  respectively. 

The  coefficients  of  wl2  and  pi2  for  the  maximum  positive 
and  negative  moments  for  the  two  beams  are  as  follows : 

Maximum 

near  Center  Maximum 

of  Span.  at  Support. 

Beams  of  two  spans  (Fig.  77) : 

Dead  load 070  .125 

Live  load 095  .125 

Beam  of  three  spans  (Fig.  72) : 

f  1st  span 080  1 

Dead  load    {  _ ,  .100 

[2d  span 025  j 

f  1st  span 100  1 

Live  load     ^  j.  .117 

i  2d  span 0/5  j 


§  155.] 


STRESSES   IN    CONTINUOUS  BEAMS. 


303 


0.2 


0.1 


0.1 


\ 


0.01 


0.0 


0.1 


0.1 


\/ 


0.2 


0.2 


.Deatf  Load 
.LLvejLoad 
_  Total  Load 


FIG.  71. — Moments  in  Beams  of  Two  Spans. 


0.3 


0.1 


0.1 


0.0 


0.1 


0.1 


V 


0.2 


0.2 


Load 


TVvfol  Trnari 

FIG.  72.—  Moments  in  Beams  of  Three  Spans. 


304  BUILDING    CONSTRUCTION.  [Cn.  VII. 

If  the  dead  arid  live  loads  are  combined  into  a  single  unit 
for  the  purposes  of  calculation,  the  proper  coefficient  for 
(w  +  p)  will  depend  on  the  relation  of  dead  and  live  load.  If, 
for  example,  the  dead  load  is  one-half  the  live  load,  then  there 
results,  for  the  first  case, 

Maximum  positive  moment  =  .087 (w  +  p)l2, 
Maximum  negative  moment  —  $(w  +  p)l2, 

and  for  the  second  case 

Maximum  positive  moment  =  .093  (w  +  p)l2, 
Maximum  negative  moment  =  .lll(w  + p)l2. 

In  Figs.  71  and  72  the  full  lines  represent  the  maximum 
moments  throughout  the  beam  for  the  condition  that  w=^p'} 
these  lines  are  particularly  useful  in  showing  the  relative  dis- 
tances from  the  supports  over  which  positive  and  negative 
moments  may  occur. 

Moments  in  Beams  of  Several  Spans. — In  the  case  of  several 
spans  it  will  be  practically  correct  in  calculating  positive 


moments  to  consider  that  the  maximum  moment  at  the  center 
of  the  span  is  the  maximum  desired.  (Strictly  the  maximum 
is  generally  not  quite  at  the  center.)  The  loading  required 
for  maximum  live-load  moments  is  illustrated  in  Fig.  72a, 
which  shows  in  (a)  the  loading  for  maximum  positive  moment 
in  spans  2-3,  4-5,  and  6-7,  and  in  (6)  the  loading  for  maximum 
negative  moment  at  support  No.  3.  For  the  former  case  each 
alternate  span  is  loaded  and,  for  the  latter,  the  two  adjoining 
spans  are  loaded  and  then  each  alternate  span.  Calculations 
have  been  made  for  each  span  and  each  support  for  4,  5,  6, 
and  7  spans,  and  given  in  Table  21  A.  It  is  found  in  general 


§  155] 


STRESSES  IN    CONTINUOUS    BEAMS. 


305 


that  for  all  spans  and  supports,  except  the  end  span  and 
support  adjacent  thereto,  the  maximum  positive  and  negative 
moments  do  not  vary  greatly  for  the  different  spans,  but  for 
these  end  spans  and  supports  they  are  considerably  larger 
than  for  intermediate  spans.  The  results  are,  accordingly, 
arranged  in  two  groups  in  the  table.  For  the  intermediate 
spans  the  greatest  value  for  the  several  spans  is  given.  The 
table  also  includes  the  previous  results  for  two  and  three  spans. 

TABLE  No.  21A. 
COEFFCIENTS  FOR  MAXIMUM  MOMENTS  IN  CONTINUOUS  BEAMS. 


Intermediate  Spans  and  Supports. 

End  Span  and  2d  Support. 

No.  of 
Spans. 

At  Center  (  +  ) 

At  Support  (  —  ) 

At  Center  (  4-  ) 

At  Support  (-) 

Dead. 

Live. 

Dead. 

Live. 

Dead. 

Live. 

Dead. 

Live. 

Two  .... 

.070 

.095 

.125 

.125 

Three  .  .  . 

.025 

.075 

.080 

.100 

.100 

.117 

Four  .  .  . 

.036 

.081 

.071 

.107 

.071 

.098 

.107 

.120 

(.115) 

Five  .... 

.046 

.086 

.079 

.111 

.072 

.099 

.105 

.120 

(.106) 

(.116) 

Six  

.043 

.084 

.086 

.116 

.072 

.099 

.106 

.120 

(.106) 

(.116) 

Seven  .  . 

.044 

.084 

.085 

.114 

.072 

.099 

.106 

.120 

(.106) 

(-116) 

The  quantities  in  parentheses  are  the  coefficients  for  live- 
load  moments  over  supports  where  the  two  adjoining  spans 
only  are  loaded.  The  effect  of  loading  each  alternate  span  in 
addition  to  these  two  spans  is  seen  to  be  small;  and  consider- 
ing that  such  a  loading  would  be  extremely  improbable,  and 
also  the  fact  that  a  comparatively  small  amount  of  load  on 
the  other  spans  would  neutralize  this  effect,  it  is  apparent 
that  the  quantities  in  parentheses  may  be  taken  as  reason- 
able maximum  values.  The  twro-span  beam  should  preferably 
be  treated  as  a  special  case. 


306 


BUILDING  CONSTRUCTION. 


[Cn.  VII. 


Finally,  leaving  out  of  account  the  two-span  beam,  the 
following  values  may  be  taken  as  reasonable  maximum  values 
for  beams  of  any  number  of  spans. 


Intermediate  Spans. 

End  Spans. 

At  Center. 

At  Support. 

At  Center. 

At  Support. 

Dead  load  moments        .    .    . 

.045 
.085 

.085 
.105 

.075 
.100 

.105 
.115 

Live  load  moments  . 

Practical  Working  Coefficients  for  Moments. — It  is  generally 
convenient  to  adopt  some  simple  fraction,  such  as  g,  fa,  or 
fa  for  the  coefficient  for  both  dead  and  live  loads  for  ordinary 
calculations,  and  if  the  same  coefficient  can  be  used  for  both  dead 
and  live  load  it  is  desirable  to  do  so.  The  live  load  will  gener- 
ally range  from  two  to  five  times  the  dead  load.  The  average 
coefficients  for  the  combined  loads,  for  various  ratios  of  live 
to  dead  load,  using  the  separate  values  above  given,  are  as 
follows : 


Intermediate  Spans. 

End  Spans. 

Ratio  of 

Live  :  Dead. 

At  Center. 

At  Support. 

At  Center. 

At  Support 

2:1 

.072 

.098 

.092 

.112 

3:1 

.075 

.100 

.094 

.112 

4:1 

.077 

.101 

.095 

.113 

5:1 

.078 

.102 

.096 

.113 

It  will  be  seen  from  this  table  that  for  ordinary  proportions 
a  single  coefficient  may  well  be  used  for  both  dead  and  live 
loads. 

Before  adopting  final  values  consideration  should  be  given 
to  certain  modifying  influences.  The  beams  and  slabs  are  not 
freely  supported  as  assumed,  but  are,  to  a  considerable  extent, 
fixed  at  the  supports.  This  tends  to  reduce  the  maximum 


§  155.]  STRESSES   IN    CONTINUOUS   BEAMS.  307 

moments.  The  supports,  also,  are  of  considerable  width,  so 
that  if  the  span  lengths  be  taken  center  to  center,  the  negative 
moment  at  the  edge  of  the  support  is  considerably  less  than 
the  caculated  maximum.  Thus  if  the  width  of  support  is  ^th 
the  span  length  the  negative  moment  at  edge  of  support  is 
about  25%  less  than  at  center  of  support.  The  slab  also  is 
greatly  strengthened  by  the  adjoining  floor  structure,  as  ex- 
plained in  Art.  156,  and  is  also  much  simpler  in  design  than 
the  beam,  and  hence  need  not  be  so  liberally  proportioned. 
Furthermore,  it  is  generally  convenient  to  use  the  same  amount 
of  steel  over  the  support  as  at  the  center,  so  that  the  moment 
at  support  will  govern  the  design.  Considering  all  these 
elements  the  following  coefficients  are  proposed  for  both  dead 
and  live  loads,  and  for  both  positive  and  negative  moments : 

For  slabs  of  medium  or  short  span : 

Intermediate  arid  end  spans fa 

For  beams  and  for  slabs  of  long  span : 

Intermediate  spans fa 

End  spans fa 

A  value  of  fa  is  commonly  used  throughout,  but  this  is 
unnecessarily  large  for  most  slab  spans,  although  it  might  well 
be  used  for  beams  if  it  is  desired  to  have  a  fixed  value  for  all 
spans. 

Remarks. — The  foregoing  calculations  assume  uniform 
moment  of  inertia  and  therefore  that  about  the  same  amount 
of  steel  is  used  for  negative  as  for  positive  moments.  The 
effect  of  variation  in  moment  of  inertia  is  discussed  in  Art.  166. 
Equal  spans  are  also  assumed.  Where  span  lengths  vary 
greatly,  special  calculations  should  be  made,  but  the  use  of 
fa  for  the  general  coefficient  will  provide  ample  strength  in 
all  ordinary  cases.  This  would  require  an  actual  resisting 
moment  at  each  support  of  only  about  fapl2  in  order  that  the 


308  BUILDING    CONSTRUCTION.  [Cn.  VII. 

center  moment   be   reduced   to   fopl2.      Cases   of  heavy   con- 
centrated loads  must  be  given  special  consideration. 

Shears.  The  maximum .  shears  near  supports  are  not 
greatly  affected  by  moving  loads.  For  intermediate  spans 
the  maximum  end  shear  may  be  taken  at  one-half  of  the  span 
load;  for  end  spans  the  shear  near  the  second  support  will 
be  approximately  six-tenths  of  a  span  load. 

156.  Effect  of  Rigid  Supports  on  the  Resisting  Moment. 
If  a  flat  slab  is  held  between  unyielding  supports,  such  as  fixed 
I-beams,  a  strength,  or  resisting  moment,  will  be  developed 
in  the  slab  even  though  there  be  no  steel  reinforcement.     Fail- 
ure cannot  take  place  without  the  crushing  of  the  concrete 
either  at  the  center  or  at  the  support.     For  short  spans  this 
resisting  moment  (the  so-called  "  arch  action  ")  is  about  as 
great  as  will  exist  in  the  slab  if  reinforced  and  simply  sup- 
ported at  the  ends.     In  the  case  of  a  flat  reinforced  slab  such 
rigid  supports  likewise  add  considerably  to  the  strength  of 
the  slab,  giving  the  effect  of  partial  continuity. 
i       In  practice,  the  supports  of  slabs  of  short  span  length, 
whether  consisting  of  I-beams  or  of  concrete  beams  of  which 
the  slab  is  a  part,  are  rendered  very  rigid  by  reason  of  the 
action  of  the   adjoining  floor-panels.     Even  where  the  slabs 
are  simply  supported  on  the  tops  of  steel  beams  the  adjoining 
slabs  prevent  to  some  extent  lateral  motion,  rendering  all  such 
spans  partially  continuous.    The  strengthening  effect  of  rigid 
supports  is,  therefore,  especially  great  in  the  case  of  narrow 
floor-spans  and  where  there  is  a  large  number  of  consecutive 
unbroken  panels.     Under  such  conditions  reinforcement  against 
negative  moment  is  hardly  necessary.    For  long  spans  and  for 
spans  on  the  outside  of  a  system  the  effect  is  small. 

157.  Slabs  Reinforced  in  Two  Directions. — If  the  panel 
between  beams  is  square,  or  nearly  so,  the  slab  may  advan- 
tageously be  reinforced  in  both  directions.  The  exact  analysis 
of  stresses  in  such  a  case  is  difficult,  if  not  impossible,  as  the 
effect  of  the  more  or  less  rigid  supports  is  especially  important 
and  the  problem  is  otherwise  difficult  of  exact  treatment. 


§  158.]  SQUARE  SLABS.  309 

The  following  solution  for  square  and  rectangular  slabs  will 
serve  to  show,  approximately,  the  relation  of  the  loads  car- 
ried by  the  two  systems  of  reinforcement.  The  results  are 
certainly  safe  and  do  not  vary  much  from  rules  of  practice, 
but  point  to  a  somewhat  more  economical  use  ^of  material. 

158.  Square  Slabs. — In  this  case  the  reinforcement  should 
be  of  equal  amount  in  the  two  directions.  It  may  be  calcu- 
lated on  the  assumption  that  one  half  the  load  is  carried  by 
each  system  of  reinforcement.  The  concrete  is  proportioned 
for  only  one  system,  or  one- half  the  load,  as  the  stresses  due 
to  the  two  systems  are  at  right  angles  to  each  other  and  it  is 
assumed  that  the  stresses  in  one  direction  do  not  weaken  the 
concrete  with  respect  to  stresses  in  the  other  direction.  The 
loading  on  each  system  is  usually  assumed  to  be  uniformly 
distributed,  resulting  in  an  equal  spacing  of  rods  throughout 
the  beam.  This  assumption  is,  however,  far  from  the  truth, 
and  while  giving  safe  results  it  is  desirable  to  consider  a  more 
exact  analysis  of  the  problem  which  will  showr  that  the  rods 
should  be  spaced  closer  at  the  center  than  at  the  edge. 

In  Fig  73,  ABCD  represents  a  square  slab  supported  on 
all  sides  and  loaded  with  a  uniform  load  w  per  unit  area. 
Consider  the  relative  amounts  of  load  carried  by  the  system 
parallel  to  aa'  and  the  system  parallel  to  mm'.  At  the  centre 
0,  and  at  all  points  on  the  diagonal  lines  AD  and  CB,  it  fol- 
lows from  symmetry  that  the  loading  is  equally  distributed  on 
the  two  systems  and  is  equal  to  w/2.  At  point  E  the  pro- 
portion of  the  load  carried  by  the  system  aa'  will  be  much 
greater  than  that  carried  by  the  system  mm',  since  for  given 
loads  the  beam  element  along  aa'  will  deflect  much  less  at 
point  E  than  will  the  element  along  mm' .  In  general,  there- 
fore, as  we  approach  the  support  BD  the  proportion  of  load 
carried  by  the  system  aa'  increases,  reaching  a  value  of  w  at 
the  extreme  end  a'.  The  distribution  of  load  on  aa'  may  then 
be  roughly  represented  by  the  ordinates  from  AB  to  the  curved 
line  aOa'  of  Fig.  74.  Consider  now  the  load  along  a  line  W. 
At  points  F  and  F'  the  load  will  be  w/2;  at  point  G'  it  will 


310 


BUILDING    CONSTRUCTION. 


[Cn.  VII. 


be  less  than  w/2,  being  the  same  as  the  load  on  the  system  mm' 
at  G.  It  will  be  shown  in  Fig.  74  by  the  ordinate  GH  from 
Oaf  to  the  line  aa'.  At  the  end  bf  the  load  will  be  w.  The 
curve  of  distribution  will  then  be  somewhat  as  represented 
by  the  line  oGV  in  Fig.  74,  in  which  G'K=GH. 

Assuming  the  curve  aOa'  to  be  a  parabola  it  is  found  that  the 
centre  bending  moment  along  the  line  aa',  for  a  beam  one 
foot  wide,  will  be  }s  (w/2)P  instead  of  &  (w/2)l2,  as  results 


i                                   H           a' 

xs 

±±^  0 

~X7 

i 

^T'"    i 

i 

2 

A 

K 

B 

FIG.  73. 


FIG.  74. 


from  the  usual  assumption.  The  spacing  of  the  rods  at  the 
centre  may  then  be  determined  on  this  basis.  At  points  inter- 
mediate between  the  centre  and  the  edge,  the  rods  might  well 
be  spaced  so  that  the  number  per  foot  would  vary  from  the 
required  number  at  the  centre  to  zero  at  the  edge,  following 
the  law  of  the  parabola.  If  N  represents  the  total  number 
required  on  the  ordinary  assumption  of  equal  spacing,  then 
|A7X|,  or  &/V,  would  represent  the  more  correct  number 
when  spaced  as  here  calculated.  Practically  as  good  results 
will  be  secured  if  the  rods  are  spaced  uniformly  at  the  usual 
spacing,  determined  by  the  formula  M=J(u'/2)Z2,  for  the  cen- 
tre half  of  the  slab,  then  gradually  reduce  the  number  per 
foot  to  the  edge  of  the  slab,  using  one-half  as  many  rods  for 
the  remaining  two  quarters.  The  total  number  used  would 
then  be  %N  instead  of  IN  as  above  determined,  but  the 
strength  would  be  ample.  If  the  slabs  are  continuous,  then 


§  159. 


RECTANGULAR    SLABS. 


311 


-a' 


FIG.  75. 


5^5   or  ^  should  be  substituted  for  I  in  the  formula  for  M,  as 
may  be  permissible. 

159.  Rectangular  Slabs  of  Greater  Length  than  Breadth. — 
As  a  slab  becomes  oblong  in  form  the  relative  amount  of  load 
carried  by  the  longitudinal  system 
becomes  rapidly  less.  Fig.  75 
represents  an  oblong  slab  of  length 
I  and  breadth  b.  Consider  a  cen- 
tral strip  one  foot  wide  along  the 
line  aa'  and  also  along  the  line  mm'. 
Suppose  the  rods  to  be  spaced 
equally  in  the  two  directions  so  that  the  moments  of  inertia 
of  the  strips  are  equal.  Let  w  =  load  per  foot  on  the  strip 
mm'  and  w'  =  load  per  foot  on  aa'.  The  deflection  of  a  beam 
uniformly  loaded  is  proportional  to  wl4,  hence,  since  the  deflec- 
tions of  the  two  beams  are  equal,  we  have  wl4  =  w'b4  or  w:uf 
=  64:/4.  That  is,  the  amount  of  the  load  carried  (per  square 
foot)  by  the  two  systems  is  inversely  proportional  to  the 
fourth  power  of  the  respective  dimensions.  For  points  nearer 
the  ends  of  the  slab  the  proportion  carried  by  the  longitu- 
dinal system  will  be  greater,  but  in  any  case  the  longitudinal 
rods  will  be  much  under-stressed. 

In  accordance  with  this  theory  the  proportion  of  the  total 
load  carried  by  the  transverse  system  for  various  ratios  of 
I: b  is  as  follows: 


Ratio  I  •  b 

1 

1   i 

1  2 

1  3 

1  4 

1  5 

Proportion   of  load   carried   by 

transverse  system  

0.50 

0.59 

0.67 

0.75 

0.80 

0.83 

In  the  report  of  the  French  Commission  the  factors  recom- 
mended give  greater  weight  to  the  strengthening  effect  of  a 
double  system.  They  are  as  follows: 


Ratio  I  •  b 

1 

1  i 

1  2 

1  3 

1  4 

1  5 

Proportion  of  load  carried  by 
transverse  system  

0.33 

0.42 

0.50 

0.58 

0.65 

0.72 

Proportion  of  load  carried  by 
longitudinal  system  

0.33 

0.26 

0.20 

0.15 

0.12 

0.08 

312  BUILDING    CONSTRUCTION.  [Cn.  VII. 

It  is  evident  in  any  case  that  if  the  length  is  as  much  as 
25%  more  than  the  breadth  the  working  stresses  in  the  longi- 
tudinal rods  will  be  much  less  than  in  the  transverse  rods,  and 
that,  in  general,  it  is  not  economical  to  reinforce  long  and 
narrow  panels  in  two  directions. 

From  this  discussion  it  is  evident  that  longitudinal  rein- 
forcement should  not  be  used  to  carry  load  in  oblong  panels 
where  the  length  exceeds  the  breadth  by  more  than  15  to  20  %. 
An  excess  of  25%  would  seem  to  be  about  the  practical  limit. 
Whatever  steel  is  placed  in  the  longitudinal  direction  is  used 
uneconomically. 

1 60.  Reinforcement  to  Prevent  Cracks. — While  longitudinal 
reinforcement  is  of  little  value  in  carrying  loads,  a  small  amount 
is  nevertheless  often  desirable   in   preventing   cracks  and  in 
binding   the   entire   structure  together.       For    a    close  beam 
spacing  such  reinforcement  is  hardly  necessaiy,  as  the  beam 
reinforcement  itself  thoroughly  ties  the  structure  longitudinally 
along  the  beam  lines.     For  wide  beam  spacing  it  is  more  im- 
portant.    Just  what  amount  of  steel  is  needed  is  a  matter  of 
experience.     The  use  of   J-inch  or  |-inch  rods  spaced   about 
two  feet  apart  is  common  practice.     If  a  metal  fabric  is  used 
for  oblong  panels,  the  longitudinal  metal  should   be  propor- 
tioned in  accordance  with  the  principles  discussed  in  this  and 
the  preceding  articles. 

161.  Floor-slabs  Supported  on  Steel  Beams.— Many  "  sys- 
tems" have  been  developed  of  this  type  of  construction,  differ- 
ing from  each  other  in  form  of  steel  used,  position  of  the  con- 
crete relative  to  the  beam,  use  of  curved  or  flat  slabs,  use  of 
various  kinds  of  hollow  tile  in  connection  with  the  concrete, 
etc.      Sufficient    examples    only  will    be  given  to  illustrate 
the  principles  involved ;  further  information  regarding  the  many 
systems  can  readily  be  had  from  trade  catalogues. 

Fig.  76  shows  the  floor  placed  directly  on  the  tops  of  the 
beams.  The  reinforcement  may  be  small  rods  or  a  mesh- 
work  of  expanded  metal  or  woven  fabric.  If  reinforced  as 
shown,  the  slab  must  be  calculated  as  a  simple  beam,  there 


§  162.] 


FLOOR-SLABS. 


313 


being  no  reinforcement  against  negative  moment  over  the  sup- 
port. For  spans  of  considerable  length  some  reinforcement 
for  negative  moments  is  desirable  to  secure  economy  and  to 


FIG.  76. 


prevent  cracks  in  the  upper  surface,  although  the  lateral 
rigidity  due  to  adjoining  panels  is  of  much  assistance,  as 
explained  in  Art.  156. 

Fig.  77  represents  a    slab  constructed  after  Hennebique's 


FIG.  77. 

system,  to  be  supported  by  walls  or  steel  beams.  Small  rods 
are  used  for  reinforcement,  every  alternate  rod  being  bent  up 
and  stirrups  of  flat  steel  looped  on  the  straight  rods.  This 
is  a  very  effective  design  to  secure  strength  against  shear  or 
diagonal  tensile  stresses,  but  except  where  the  floor-load  is 
very  heavy  special  shear  reinforcement  is  hardly  needed  in 
floor-slabs.  Fig.  78  shows  a  more  common  design  of  non- 


FIG.  78. 

continuous  slab,  the    concrete  being  supported  on  the  lower 
flange  and  the  entire  beam  surrounded. 

Fig.  79  shows  a  standard  form  of  construction   in  which 


•-    V: -;.'-'.• 


FIG.  79. 

the  slab  is  practically  continuous.     The  reinforcing  material 
may  be  rods  or  a  metal  fabric  continuous  over  several  spans. 


314 


BUILDING  CONSTRUCTION. 


[Cn.  VII. 


Figs.  80  and  81  show  two  forms  in  which  a  bar  is  hooked 
around  the  beam  flange. 


FIG.  80. 


FIG,  81. 

Many  other  forms  are  employed,  some  using  a  concrete  arch 
with  more  or  less  reinforcement.  In  some,  also,  the  concrete 
slab  is  brought  down  somewhat  below  the  beam,  giving  a  plane 
surface  on  the  under  side. 

162.  Floor-slabs  in  All-concrete  Construction.— \V here  con- 
crete beams  are  used  the  slab  and  beam  are  usually  built  simul- 
taneously, giving  a  monolithic  structure.  The  slab  thus  con- 
stitutes part  of  the  beam,  but  to  be  effective  these  two  parts 
must  be  well  tied  together.  Where  cross-beams  are  used  the 
span  of  the  slab  will  commonly  range  from  4  to  6  feet  in  length. 
For  such  short  spans  a  reinforcement  of  rods  or  metal  mesh- 
work  near  the  bottom  only  will  be  effective  as  explained  in  Art. 
156.  This  reinforcement  if  of  rods,  should  be  laid  v.  ith  lapped 
and  broken  joints  to  give  continuity  and  to  prevent  the  localiza- 
tion of  contraction  cracks  in  undesirable  places  (Fig.  82).  The 


FIG.  82. 

beam,  if  well  bonded  to  the  slab,  will  make  a  very  rigid  sup- 
port comparable  to  the  I-beam. 

In  the  case  of  spans  longer  than  5  to  6  feet  it  becomes  desir- 
able to  reinforce  against  negative  moment.  This  may  readily 
be  done  by  bending  up  a  part  or  all  of  the  rods  and  extending 


§  163.] 


BEAMS   AND    GIRDERS. 


315 


the  bent  ends  beyond  the  beam.    Fig.  83  illustrates  two  arrange- 
ments of  this  kind.     In  either  case  the  amount  of  steel  at  the 


""  WZ& 


4 

4 


FIG.  83. 


top  above  the  beam  is  the  same  as  at  the  bottom  in  the  centre 
of  the  slab.  The  result  may  also  be  arrived  at  by  using  sep- 
arate straight  rods,  as  shown  in  Fig.  84.  The  plan  of  bent 


FIG.  84. 

rods  has  a  slight  advantage  as  it  reinforces  somewhat  against 
shearing  failures,  but  this  is  not  usually  important  in  slabs. 
For  very  heavy  loads,  however,  it  becomes  of  importance,  and 
the  same  care  should  be  used  as  in  the  design  of  large  beams. 
Fig.  85  shows  the  Hennebique  bent-rod  and  stirrup  sys- 
tem applied  to  long-span  slabs. 


FIG.  85 

The  length  of  span  over  which  negative  moment  is  likely 
to  exist  may  be  estimated  from  Fig  72.  It  is  seen  that  in 
the  centre  span  of  a  three-span  girder,  where  the  dead  load  is 
one-half  the  live  load,  negative  moment  may,  under  extreme 
conditions,  occur  entirely  across  the  beam.  For  long  spans  a 
top  reinforcement  at  least  to  the  third  point  will  be  desirable, 


316  BUILDING  CONSTRUCTION.  [Cn.  VII. 

but  for  short  spans  a  less  extensive  reinforcement  will  be 
sufficient.  The  effect  of  a  less  amount  of  steel  is  discussed 
in  Art.  165.  Other  examples  of  slab  construction  are  shown 
in  Art.  168. 

163.  Beams  and  Girders. — Econondcal  Arrangement.— The 
arrangement  of  columns,  girders,  and  beams  is  determined 
according  to  the  same  principles  as  in  steel  construction.  The 
spacing  of  columns  and  girders  will  be  determined  largely  by 
architectural  considerations.  The  best  spacing  of  cross-beams 
will  differ  in  different  cases.  Where  the  spacing  of  girders  is 
not  too  great  (12  to  15  feet)  and  where  cross-beams  are  not 
needed  to  secure  lateral  stiffness,  it  will  be  a  question  of  omit- 
ting all  cross-beams,  of  inserting  them  only  at  columns  so  as 
to  form  a  square  or  nearly  square  panel,  or  of  spacing  them 
at  closer  intervals  of  4  to  8  feet,  using  two  or  more  to  a  girder- 
panel.  The  preceding  analysis  shows  that  double  reinforce- 
ment will  not  be  economical  for  oblong  panels.  Cross-beams, 
if  used,  should  therefore  be  arranged  to  give  very  nearly  square 
panels  or  else  be  spaced  much  more  closely,  designing  the  rein- 
forcement so  as  to  carry  the  entire  load  to  the  beams  and 
thence  to  the  girders. 

If  not  otherwise  needed,  the  use  of  cross-beams  to  secure 
square  panels  effects  little  if  any  saving.  The  amount  of  con- 
crete will  be  less,  but  the  amount  of  steel  required  will  be  more, 
and  the  extra  beam  will  be  more  costly  per  unit  volume  than 
the  slab.  However,  for  the  sake  of  lateral  stiffness  it  will 
usually  be  desirable  to  place  cross-beams  at  columns. 

Where  close  spacing  of  beams  is  adopted  the  best  arrange- 
ment depends  upon  the  loading  and  the  working  stresses,  as 
well  as  upon  the;  cost  of  the  material  and  forms.  Heavy  loads 
and  low  stresses  call  for  large  weights  of  concrete  and  tend  to 
require  the  use  of  the  material  more  in  the  form  of  deep  ribs 
or  beams,  as  the  deeper  the  beam  the  greater  its  moment  of 
resistance  for  a  given  volume.  If  cross-beams  are  used,  a 
spacing  greater  than  10  or  12  feet  or  less  than  4  or  5  feet  will 
seldom  be  economical.  Architectural  considerations  will  often 


§  165.] 


BEAMS   AND    GIRDERS. 


317 


govern,  and  frequently  building  regulations  relative  to  ratio 
of  span  to  depth  will  control. 

164.  Distribution  of  Floor-loads  to  Beams. — Where  the  floor- 
slab  is  reinforced  in  one  direction  only  the  load  will  prac- 
tically all  be  transmitted  to  the  corresponding  beams,  but  at 
the  ends  of  the  panels  a  small  part  will  be  transferred  directly 
to  the  girder.  This  may  be  neglected  in  the  calculations.  In 
the  case  of  reinforcement  in  two  directions,  unless  the  panel 
is  nearly  square,  the  load  may  still  be  assumed  as  all  trans- 
ferred to  the  side  beams.  If  the  panels  are  square,  or  nearly 
so,  the  distribution  may  be  assumed  in  accordance  with  the 
discussion  of  Art.  158.  Thus  the  load  brought  to  point  a' 
(Fig.  74)  will  be  one-half  of  the  area  below  the  curve  aOa', 


FIG.  86. 

and  the  load  brought  to  &'  will  be  one-half  the  area  below  the 
curve  bG'V,  etc.  The  distribution  along  the  beam  will  then 
follow  some  such  law  as  represented  by  the  shaded  area  in 
Fig.  86  (a),  the  total  load  being  necessarily  \wl,  where  w  =  floor- 
load  per  square  foot.  It  will  be  sufficiently  accurate  to  assume 
this  curve  a  parabola.  The  centre  bending  moment  in  the 
beam,  assumed  as  a  simple  beam,  will  then  be  equal  to 


A  distribution  of  load  as  represented  in  Fig.  86  (6),  as  is  some- 
times assumed,  gives  a  centre  moment  equal  to  £$ivl2,  a  value 
about  7%  higher  than  the  above.  A  uniform  distribution  gives 
a  moment  equal  to  £%wl2,  a  value  20%  lower. 


318  BUILDING  CONSTRUCTION.  [Cn.  VII. 

165.  Design  of  Cross-beams. — In  the  design  of  beams  the 
chief  features  are  the  determination  of  the  cross-section,  the 
amount  of  steel  and  its  make-up,  provision  for  shearing  stress, 
provision  for  negative  bending  moment  and  connections  with 
slabs,  other  beams,  and  columns.  The  proportions  of  the  beam, 
whether  considered  as  a  rectangular  beam  or  as  a  T-beain, 
will  be  determined  by  considerations  discussed  in  Chapter  V. 
Ratios  of  depth  to  width  greater  than  2  or  2J  are  seldom  used. 
Requirements  of  head-room,  space  for  rods,  and  shearing 
strength  will  limit  the  possible  variations  in  proportions  to  a 
comparatively  narrow  range.  Deep  beams  are  economical  of 
concrete  but  cost  more  for  forms  than  do  shallow  beams. 

If  the  beam  may  be  calculated  as  a  T-beam,  the  width  of 
slab  wrhich  may  be  counted  on  as  a  part  of  the  beam  is  an  im- 
portant question.  Specifications  usually  allow  a  width  of  six 
to  ten  times  the  thickness  of  the  slab,  but  not  to  exceed  the 
width  betwreen  beams.  As  regards  strength  it  would  be  verj* 
difficult  to  secure  so  thorough  a  reinforcement  of  w^eb  as  to 
make  it  possible  to  crush  a  flange  as  much  as  four  times  the 
width  of  the  web;  the  excessive  shearing  stresses  in  the  web 
would  cause  failure.  As  regards  stiffness,  which  controls  the 
position  of  the  neutral  axis,  the  width  of  the  slab  to  be  counted 
as  part  of  the  beam  may  and  should  be  taken  relatively  great. 
The  width  of  flange  being  known,  the  design  of  the  T-beam 
consists  chiefly  in  the  design  of  the  web  and  the  calculation 
of  the  steel  cross-section.  It  will  be  only  in  the  case  of  large 
girders  that  the  compressive  stress  in  the  concrete  will  be  a 
determining  factor.  Usually  there  is  a  large  excess  of  material. 

If  the  beam  is  to  be  considered  as  continuous  over  sup- 
ports, the  moment  of  resistance  at  the  support  must  also  be 
investigated.  At  this  point  the  tension  side  is  uppermost  and 
the  effective  beam  is  now  a  rectangular  beam.  The  maximum 
moment  is  about  the  same  as  at  the  centre,  thus  requiring 
about  the  same  amount  of  steel  at  the  top  as  is- required  in 
the  centre  of  the  span  at  the  bottom.  The  maximum  com- 
pression in  the  concrete  will  be  greater  than  in  the  centre  and 


§  166.]  BEAMS    AND    GIRDERS.  319 

will  probably  determine  the  size  of  beam  required  unless 
special  provision  is  made  for  these  stresses.  This  may  be 
done  by  increasing  the  depth  of  the  beam  near  the  end,  as 
shown  in  Fig.  87,  or  by  the  use  of  compressive  reinforcement. 
Such  reinforcement  may  be  provided  to  a  considerable  extent 
by  merely  continuing  the  horizontal  steel  sufficiently  to  give 
the  necessary  bond  strength  (see  Art.  123).  If  the  horizontal 
steel  near  the  end  amounts  to  as  much  as  1%  of  the  rectan- 
gular section,  then  considering  both  of  two  adjoining  beams 
there  would  be  available  about  2%  of  compressive  reinforce- 
ment. By  Plate  XII,  p.  286,  this  amount  would  reduce  the 
compressive  concrete  stresses  about  42%.  This  would  usually 
be  sufficient.  Inasmuch  as  a  slight  excess  of  stress  at  this 


FIG.  87. 

point  does  not  in  any  way  endanger  the  structure,  merely 
increasing  somewhat  the  positive  moment  on  the  beam,  it 
would  seem  to  be  proper  to  permit  the  use  of  a  higher  working 
stress  than  at  the  center.  An  increase  of  10-15%  would  be 
entirely  safe.  The  necessarjr  top  steel  at  the  end  may  be 
provided,  as  in  the  slab,  by  bending  up  a  portion  of  the  lower 
rods,  or  by  using  separate  short  rods,  or  by  both  methods 
combined.  To  provide  thoroughly  for  negative  moment  the 
upper  reinforcement  should  extend  to  about  the  third  point, 
and  in  some  cases  still  farther.  Various  arrangements  of 
berit-up  rods  are  illustrated  in  the  examples  cited  in  Art. 
168. 

It  has  been  assumed  in  the  determination  of  positive  and 
negative  bending  moments  in  Art.  155  that  the  moment  of 
inertia  of  the  beam  is  uniform  throughout.  As  there  shown, 
the  resulting  *  maximum  moments  at  center  and  support  are 
not  greatly  different  and  for  all  practical  purposes  may  be 


320  BUILDING   CONSTRUCTION.  [Cn.  VII. 

taken  as  equal,  so  that  if  fully  reinforced  the  amount  of  steel 
and  the  moments  of  inertia  will  accord  with  the  assumption. 
It  is  the  practice  of  some  designers,  however,  to  consider  the 
beam  primarily  as  a  simple  beam  and  design  it  to  carry  all, 
or  nearly  all,  of  the  load  as  such.  A  relatively  small  amount 
of  steel  is  then  placed  in  the  top  of  the  beam  over  the  sup- 
port, mainly  to  prevent  objectionable  cracks,  but  which  is  also 
in  some  cases  counted  upon  to  carry  a  portion  of  the  moment. 
It  is  therefore  of  considerable  importance  to  determine  the 
actual  moments  and  stresses  which  occur  at  the  centre  and 
support  in  such  a  case. 

This  problem  has  been  concisely  analyzed  by  Mr.  P.  E. 
Stevens,*  and  the  following  results  are  from  his  paper.  He 
assumed  a  uniform  moment  of  inertia,  =/0,  for  that  portion 


M0I0 


over  the  support  in  which  negative  moments  exist,  and 
another  moment  of  inertia,  =/i,  for  that  portion  of  the  central 
part  of  the  span  in  which  positive  moments  exist  (see  Fig. 
87a).  Let  M0  =  bending  moment  at  support  and  MI  ^bend- 
ing moment  at  center.  Then  for  a  uniformly  distributed  load 
the  ratios  of  these  bending  moments  for  various  ratios  of  /Q:/I 
are  as  given  in  the  table  on  page  321. 

The  table  also  gives  in  the  third  and  fourth  columns  the 
values  of  these  moments  expressed  as  percentages  of  the  centre 
moment,  %wl2,  for  a  simple  beam.  In  the  last  column  are  given 
the  ratios  of  the  corresponding  unit  stresses  in  the  steel,  /o 
and  fiy  assuming  the  moments  of  inertia  to  be  proportional 
to  the  amount  of  steel  used.  If,  for  example,  the  moment  of 
inertia  is  the  same  throughout,  the  moment  at  the  centre 
is  X$wl2  and  at  the  end  is  |X§^2,  as  is  well  known.  The 


*  Trans.  Am.  Soc.  C.  E.,  Vol.  LX,  1908,  p.  496. 


166.] 


BEAMS    AND    GIRDERS. 


321 


MOMENTS  AND  STRESSES  IN  BEAMS  WITH  VARIABLE  MOMENT 

OF  INERTIA 


Ratio,  70:/i. 

Ratio,  Mo  :  Mi. 

Mi 

kwV 

Mo 

bwP 

Ratio  of  Stresses, 
/o  :/i. 

i 

0.8 

.55 

.45 

4.1 

i 

0.9 

.52 

.48 

3.7 

* 

1.1 

.48 

.52 

3.2 

i 

1.4 

.42 

.58 

2.8 

i 

2.0 

.33* 

.66| 

2.0 

U 

2.6 

.28 

.72 

1.7 

2 

3.0 

.25 

.75 

1.5 

6 

5.8 

.15 

.85 

1.0 

amount  of  steel  should  then  be  based  upon  the  end  moment 
and  made  uniform.  Again  suppose  the  amount  of  steel  at 
the  support  be  made  one-half  that  at  the  center  and  that  the 
center  be  designed  for  fowP=*.8X\wP*  The  actual  moment 
will  be  |2  of  the  assumed  moment  and  the  stress  at  the  center 
will  be  the  same  proportion  of  the  assumed  working  stress. 
At  the  end  the  fiber  stress  will  be  |§X2.8  =  1.5  times  the 
assumed  working  stress.  If  a  small  amount  of  steel  be  placed 
at  the  end,  such  that  /0//i  =  i,  and  the  full  bending  moment 
provided  for  at  the  center,  the  stress  at  the  center  will  be 
55%  of  the  working  stress  arid  at  the  end  will  be  4.1  X. 55  = 
2.25  times  the  working  stress. 

The  treatment  of  girders  is  the  same  as  described  for  beams, 
it  being  especially  important  that  the  reinforcement  pass  well 
through  the  column. 

The  arrangement  of  shear  or  web  reinforcement  for  beams 
and  girders  is  of  great  importance,  as  it  is  in  these  forms  where 
the  web  tensile  stresses  will  be  high.  At  points  where  the 
allowable  shearing  stress  in  the  concrete  is  exceeded  steel  must 
be  added  in  some  form  to  carry  a  part  of  the  stress,  as  explained 
in  Art.  125.  Where  bent-up  rods  are  used,  as  in  Fig.  87,  these 
rods  aid  greatly  in  carrying  shear,  and  where  not  spaced  too 
widely  may  be  counted  on  to  add  perhaps  50%  to  the  strength 


322  BUILDING  CONSTRUCTION.  [Cn.  VII. 

of  the  web.  For  thorough  web  reinforcement  the  stirrup  is 
usually  employed,  or  some  form  of  bent  bar  closely  spaced. 
This  reinforcement  may  be  calculated  as  explained  in  Art.  125,  not 
too  much  reliance  being  placed  on  one  or  two  bent  rods.  Web 
reinforcement  will  usually  be  needed  only  for  the  end  quarter 
or  third  of  the  beam.  Near  the  support,  where  the  moment  is 
negative,  the  tendency  is  for  diagonal  cracks  to  start  at  the 
top,  while  farther  along  the  cracks  tend  to  start  at  the  bottom, 
as  shown  in  Fig.  87.  Stirrups  at  points  of  negative  moment 
should  loop  about  the  upper  bars,  and  at  points  of  positive 
moment  should  loop  about  the  lower  bars.  A  correct  appre- 
ciation of  the  diagonal  stresses  in  such  continuous  beams  is 
important. 

The  beam  should  be  well  bonded  to  the  slab,  especially 
near  the  end  where  the  differential  stresses  between  the  two 
parts  are  large.  This  is  well  accomplished  by  means  of  the 
bent  rods  brought  up  as  high  as  possible,  and  by  means  of  the 
slab  reinforcement  which  crosses  the  beam.  Along  the  centre 
of  the  beam  the  matter  is  not  of  so  great  importance,  but  it 
is  better  to  provide  such  bond  by  some  form  of  vertical  rein- 


(a) 


forcement,  such  as  stirrups,  extending  up  into  the  slab  at 
occasional  intervals.  This  is  of  especial  importance  in  the  case 
of  girders  where  the  main  slab  reinforcement  runs  parallel  to 
the  beam.  A  good  bond  is  also  more  necessary  the  thinner  the 
sections.  Sections  shown  in  Fig.  88,  (a)  and  (6),  are  more 
favorable  than  such  a  section  as  in  Fig.  88  (c).  Sharp  reentrant 
angles  in  such  a  brittle  material  as  concrete  are  points  of  weak- 
ness, and  where  they  exist  a  steel  bond  is  desirable. 

1 66.  Columns. — There  is  little  to  be  said  here  relative  to 
column  design.     Much  difference  of  opinion  still  exists  as  to 


§  168.]  FLOOR    AND    COLUMN    DESIGN.  323 

the  use  of  large  or  small  quantities  of  steel  and  methods  of 
calculation.  A  conservative  course  should  be  pursued  in  this 
matter,  as  the  columns  and  beams  in  a  reinforced  structure 
are  the  vital  parts  of  the  structure.  Working  stresses  in 
columns  such  as  700  or  800  lbs/in2  should  not  be  employed. 
Where  large  areas  of  steel  are  used,  and  figured  at  ordinary 
working  stresses,  such  steel  skeleton  should  not  rely  upon 
the  concrete  for  rigidity.  Concrete  may,  however,  be  relied 
upon  to  transmit  loads  from  girders  to  columns.  Where  small 
areas  of  steel  are  used  the  rods  should  be  well  lapped  at  the 
floor-level,  and  those  from  the  lower  columns  should  extend 
upwards  the  full  depth  of  the  connecting  beams.  The  rods 
should  be  well  banded  together  by  steel  bands  or  large  wire 
so  as  to  hold  all  parts  in  position  and  to  strengthen  the  column 
circumferentially.  Unless  such  banding  is  spaced  very  closely 
it  should  not  be  counted  upon,  however,  as  " hooping." 
Brackets  under  all  connecting  girders  are  serviceable  in 
stiffening  the  frame  as  well  as  in  decreasing  the  stress  in  the 
girders.  Rods  of  connecting  girders  should  pass  well  through 
the  columns. 

167.  Eccentric  Loads  on  Columns. — Where  loads  are  applied 
on  free  brackets  or  cantilevers  the  load  is  definitely  eccentric, 
and  the  moment  clue  to  the  same  can  readily  be  calculated. 
Moments  are  also  caused  in  columns  by  unevenly  loaded  panels 
through  the  rigid  beam  connections.  Assuming  the  beams 
rigidly  fixed  at  the  ends,  a  panel  load  on  one  without  a  load 
on  the  corresponding  one  on  the  opposite  side  will  cause  a 
bending  moment  in  the  beam  at  the  column  equal  to  ^pl2, 
where  p  =  live  load  per  lineal  foot  of  beam.  This  moment  is 
resisted  mainly  by  the  column  and  the  members  attached  to 
it  in  the  same  plane  as  the  loaded  beam,  and  in  proportion  to 
their  moments  of  inertia  divided  by  their  lengths.*  If  the  two 
beams  are  about  as  rigid  as  the  column,  then  the  moment  in 
the  column  above  and  below  the  floor  will  be  about  one-fourth 
of  the  given  moment,  =--?jpl2 .  This  indicates,  roughly,  what 

*  See  Johnson's  Framed  Structures,  Art.  154. 


324 


BUILDING  CONSTRUCTION. 


[Cn.  VII. 


may  be  expected  from  unequally  loaded  floors.  In  the  lower 
stories  of  a  high  building  such  a  moment  would  be  of  little 
consequence,  but  in  the  upper  floors  it  might  add  a  large  per- 
centage to  'the  column  stress. 

1 68.  Examples  of  Floor  and  Column  Design. — The  follow- 
ing examples  have  been  selected  from  published  designs  as 
representing  good  practice  and  as  illustrating  more  or  less 
specifically  various  features  of  design. 


N&o&i        •* 

|Xi8  ''Bars  8  apart,  alternate  bars 


i 


'••:> 


, 
bent  upward  as  shown 

SECTION.,  BEAMS 


pp£f 

m 

k&vcj 

- 

•    r  • 

"H 

>i  12"  [t 

,4°  Loops                                                                                    ^  Bars 
ELEVATION,  BEAMS 

-v^_ 

j 

} 

i 

.  ! 

pp 

rrr 

i: 

/-\      i  """"•j-^  IMl                              1^  ^4-""^  \,^-T\ 

r- 

"Ill   ' 

-  ^  L' 

ELEVATION;  GIRDERS 


Two  ?4    bars 

,      >.      JT  j_  ; 

Hoops-^jNTV  ~~_{ 
4"apart~l  |-V /H 


FIG.  89.  —  Details  of  the  Robert  Gair  Factory,  Brooklyn. 

Fig.  89  illustrates  the  details  of  the  Robert  Gair  factory, 
Brooklyn.*  The  reinforcement  of  the  columns  varies  from 
eight  lj-m-  round  rods  at  the  base  to  four  f-in.  rods  at  the 
top.  In  the  lower  stories  the  bars  are  threaded  and  connected 
by  sleeves.  The  rods  are  connected  by  hoops  spaced  from 
4  to  10  in.  apart.  The  girders  are  about  16  ft.  apart,  and  the 


*  Eng.  Record,  Vol.  51,  1905,  p.  279. 


168.] 


FLOOR  AND  COLUMN  DESIGN. 


325 


beams  about  one-third  of  this  distance.  Features  of  design 
to  be  noted  are  the  brackets  on  the  columns,  bent  rods  and 
stirrups  in  the  beams,  bent  rods  in  the  slabs,  and  longitudinal 
reinforcement  by  the  use  of  ^-inch  bars.  The  stirrups  are 
rather  widely  spacea.  The  Ransome  bar  was  used  except  in 
the  columns. 

Fig.  90  shows  the  details  of  the  Park  Square  Garage,  Bos- 
ton.*   The  slabs  are  reinforced  with  expanded  metal  brought 

i-J/Rods 
J/Ties- 


7  SRods^Diam. 

*2-J/ Rods  *  Rods  l"Diam 

GIRDER 


SECTION  A-A 


/2  Rods  I'Diam. 


m 


2  Rods  1  Diam.-^  ^3  Rods  1  Diam. 

23'0"  for  Front  Beams 


23^'for  Garage  Beams 
BEAM 

FIG.  90.— Details  of  the  Park  Square  Garage,  Boston. 


near  to  the  top  surface  over  the  beams.  They  were  calcu- 
lated as  continuous  girders.  Beams  and  girders  were  cal- 
culated as  T  sections  and  figured  on  the  basis  of  375  lbs/in2 
compressive  stress,  and  30  lbs/in2  shearing  stress  with  no 
web  reinforcement,  and  100  lbs/in2  with  such  reinforcement. 
The  column  reinforcement  consisted  of  round  rods  from  2|  in. 
to  |  in.  in  diameter.  They  were  banded  by  J  in.  bands.  The 
concrete  used  in  the  columns  was  1:1J:3;  for  the  lower  parts 
of  the  beams,  1 :2J:4;  and  for  the  upper  parts  of  the  beams  and 
for  slabs  1:2^:5.  The  close  spacing  of  the  stirrups  is  note- 
worthy. 


*  Eng.  Record,  Vol.  52,  1905,  p.  373. 


326 


BUILDING  CONSTRUCTION. 


[Cn.  VII. 


Fig.  91  shows  the  construction  for  the  Thompson  arid  Mor- 
ris factory,  Brooklyn.     Corrugated  bars  are  used  throughout. 


3  of  which  bend  up 


GIRDER  DETAILS 

FIG.  91.  —  Details  of  the  Thompson  and  Morris  Factory,  Brooklyn. 

The  girders  are  spaced  12  to  15  ft.  apart  and  the  beams  3  ft. 
9  in.  apart,  four  to  a  panel.  The  heavy  reinforcement  for 
negative  moment  should  be  noted. 

Fig.  92  shows  details  of  the  Citizens'  National  Bank  Build- 
ing, Los  Angeles,  Cal.*  Large  girders  connect  the  columns  in 
both  directions,  forming  panels  17  ft.  by  22  ft.  These  panels 
are  then  subdivided  into  four  smaller  ones  by  cross-beams  in 
both  directions,  a  somewhat  peculiar  arrangement  used  prob 
ably  for  architectural  effect.  The  slabs  are  reinforced  both 


PfCo  92. — Details  of  the  Citizens*  National  Bank  Building,  Los  Angeles. 

ways  by  f-in.  twisted  bars,  4J  in.  apart.     The  stirrups  are  flat 
bands  :n//X2"  and  spaced  18  ins.  apart  except  near  the  end, 
as  shown.    They  are  looped  about  the  rods  in  a  very  effective 
*  Eng.  News,  Vol.  56,  1906,  p.  16. 


168.1 


FLOOR  AND  COLUMN  DESIGN. 


327 


manner.     Note  the  sleeve-splice  for  the   column  bars.     The 
beams  and  columns  are  made  of  1:2:3  concrete. 

Fig.  93  shows  a  typical  girder  constructed  with  the  Kahn 
bar  illustrated  in  Fig.  7,  Art.  33.  By  using  inverted  bars  over 
the  supports  negative  moment  can  be  provided  for,  and  at  the 
same  time  additional  shear  reinforcement. 


FIG.  93. — Reinforcement  with  the  Kahn  Bar. 

In  executing  work  a  practical  difficulty  of  considerable 
importance  is  that  of  placing  and  keeping  all  bars  in  their 
proper  position  until  the  concrete  is  in  place.  Very  consider- 
able labor  is  required  in  wiring  bars  in  position,  or  in  provid- 
ing other  means  of  support,  and  careful  supervision  is  necessary 
during  construction  to  see  that  they  remain  in  place.  To 
avoid  these  difficulties  various  arrangements  have  been  devised 
for  fastening  together  all,  or  a  part,  of  the  rods  of  a  single  span 
into  a  group  which  can  be  handled  as  a  unit,  giving  rise  to  the 
so-called  "unit  frame".  These  units  are  obviously  not  so 
adaptable  to  a  great  variety  of  conditions  as  single  independent 
bars,  but  their  advantages  are  considerable  and  they  are  being 
used  to  quite  an  extent.  Fig.  94  illustrates  one  such  type  of 


FIG.  94. — Unit  Frame. 


construction  manufactured  by  the  Unit  Concrete  Steel  Frame 
Co.  of  Philadelphia,  and  has  been  used  in  several  buildings. 
Some  of  the  transverse  slab  rods  pass  through  the  upper  ends 
of  the  stirrups  as  shown. 


328 


BUILDING  CONSTRUCTION. 


[Cn.  VII. 


Fig.  95  illustrates  a  kind  of  unit  reinforcement  on  the  Bertine 
system  and  used  in  the  warehouse  of  the  Bush  Terminal  Co., 
Brooklyn.*  Round  rods  are  used  and  tied  together  by  round 
steel  stirrups. 


1J46  2-1/16 

FIG.  95. — Unit  Frame. 

In  all  the  examples  here  given  it  is  to  be  noted  that  the 
differences  of  detail  refer  almost  entirely  to  the  method  of 
caring  for  the  shearing  stresses  and  in  handling  the  reinforcing 
members.  The  beam  and  slab  arrangement  is  used  in  all. 

A  system  of  construction  quite  radically  different  from  the 
foregoing  is  shown  in  Fig.  96,  called  the  "  mushroom  "  system, 
devised  by  Mr.  C.  A.  P.  Turner.  No  beams  or  ribs  are  used, 
the  loads  being  transmitted  from  floor-slab  directly  to  the 
column.  The  reinforcement  is  essentially  radial  and  the  column 
is  enlarged  at  the  top  to  increase  the  circumference  at  the  line 
of  maximum  stress  in  the  slab.  The  floor  is  of  uniform  thick- 
ness throughout. 

To  a  certain  extent  this  type  of  construction  follows  the 
natural  lines  of  stress  more  closely  than  the  rectangular  ribbed- 
panel  type;  it  is  best  adapted  to  large  areas  with  few  large 
openings. 

The  analysis  of  stresses  in  this  system  may  be  made 
approximately  by  the  application  of  the  method  given  in  Art. 
150,  Chapter  VI,  and  Plates  X  and  XI.  In  applying  this 
method  to  a  continuous  floor  like  the  "  mushroom  "•  system, 
an  estimate  must  first  be  made  of  the  position  of  the  "  line  " 
of  inflexion  with  reference  to  the  column.  Noting  that  the 
point  of  inflexion  of  a  beam  fixed  at  the  ends  and  uniformly 

*  Eng.  Record,  Vol.  53,  1906,  p.  36. 


§  168.] 


FLOOR  AND  COLUMN  DESIGN. 


329 


loaded  is  about  one-fifth  the  span  length  from  the  end,  a  suf- 
ficiently close  estimate  of  the  "line"  of  inflexion  can  be  made. 
It  will  evidently  be  nearer  the  column  than  if  the  support  were 
a  continuous  wall.  Having  estimated  the  line  of  inflexion  the 
area  within  may  be  treated  roughly  as  a  circular  plate  loaded 
with  the  given  uniform  load  on  its  area  and  a  vertical  load 
along  its  periphery  equal  to  the  remaining  part  of  the  load 


TOP  VIEW  OF  COLUMN  FRAMING 


%  Round  Rods,  C  C.  to  C. 
PLAN  SHOWING  FLOOR  REINFORCEMENT 

FIG.  96.— The 


SECTIONAL  ELEVATION 

Mushroom"  System. 


tributary  to  the  column.  The  diagrams  then  apply  directly. 
Thus,  suppose  the  columns  are  spaced  16  ft.  apart  and  are  20 
ins.  in  diameter  at  their  upper  ends.  Suppose  the  load  to  be 
150  lbs/ft2  over  the  entire  area.  With  columns  16  ft.  apart,  the 
diagonal  spacing  will  be  about  22.5  ft.  The  line  of  inflexion 
will  probably  not  be  less  than  3  ft.  nor  more  than  4  ft.  from 
the  column  centre.  Call  it  3.5  ft.  The  area  of  this  circle  will 


330  BUILDING  CONSTRUCTION.  [Ce.  VII. 

be  38.5  sq.  ft.  The  area  of  the  entire  square  tributary  to  the 
column  is  16X16  =  256  sq.  ft.  Hence  the  total  load  applied 
along  the  periphery  will  be  (256 -  38.5)  X 150  =  32,600  Ibs., 
which  will  be  equal  to  1480  Ibs.  per  lineal  ft.  From  Plates 
X  and  XI  the  value  of  MI  and  M2  are  found  to  be :  (a)  for 
the  direct  load  of  150  lbs/ft2,  ^  =  280  ft.-lbs.,  M2  =  1040  ft.-lbs.; 
(6)  for  the  peripheral  load  of  1480  lbs/ft,  Mi  =  2960  ft.-lbs., 
M2=8650  ft.-lbs.  If  TI  had  been  assumed  at  4  ft.  the  values 
of  M2  would  have  been  1500  and  9180  ft.-lbs.,  respectively. 
An  increase  in  column  diameter  to  30  ins.  would  reduce  the 
moments  M2  to  about  980  and  6600  ft.-lbs.  respectively,  as- 
suming a  value  of  ri  of  4  ft. 

In  the  illustrations  shown  the  columns  have  been  mainly 
reinforced  by  longitudinal  rods.  Various  types  of  banded  or 
hooped  columns  are  used  more  or  less,  but  usually  in  connec- 
tion with  longitudinal  reinforcement.  From  the  discussion  of 
Chapter  IV  it  would  seem  that  large  amounts  of  hooped  rein- 
forcement should  not  be  counted  upon  too  greatly  in  the  strength 
of  the  column.  In  some  forms  the  columns  are  banded  with 
spirally  wound  hooping,  as  in  the  Considere  column,  in  others 
flat  steel  is  used  in  riveted  or  welded  hoops.  Expanded  metal 
is  also  used  by  wrapping  around  longitudinal  bars. 

169.  Footings. — The  problem  of  the  design  of  footings  is 
in  general  the  same  as  that  of  floors.  On  account  of  the  heavy 
concentrated  loads  and  the  large  unit  upward  pressures  of  the 
earth  against  the  footings  the  beam  construction  will  be  rela- 
tively heavy.  The  beams  will  be  short  and  deep  and  will 
require  special  attention  to  provide  against  excessive  shearing 
and  bond  stresses.  For  single  footings  of  ordinary  size  a  single 
symmetrical. slab  is  most  convenient.  For  larger  footings  and 
for  footings  carrying  more  than  one  column,  a  combination  of 
beam  and  slab,  similar  to  floor  construction,  is  often  most 
economical. 

It  is  difficult  to  calculate  accurately  the  stresses  in  a  square 
footing,  but  assumptions  may  be  made  which  will  simplify 
the  problem  and  give  results  well  on  the  safe  side  (see  Fig.  97). 


169.] 


FOOTINGS. 


331 


As  a  general  principle  the  pressures  should  be  carried  as  di- 
rectly as  possible  from  the  extremities  to  the  centre.  Two 
sets  of  main  reinforcing  rods  aaf  and  W  will  then  be  used 
as  shown  in  the  figure.  The  reinforcing  of  the  remaining  cor- 
ners can  best  be  done  by  sets  of  diagonal  rods  dd' '.  If  these 
cannot  cover  the  area,  then  a  few  short  cross-rods  may  be 
used.  Reinforced  in  this  way  the  total  pressure  on  the  area 
A  BCD  may  be  assumed  to  be  carried  to  the  line  BC,  where 
the  bending  moment  and  shear  will  be  a  maximum.  Figured 
as  a  free  cantilever  the  resulting  stresses  will  be  higher  than 
actually  exist.  If  the  entire  square  be  reinforced  by  rods  in 


e  d 


d'     A 


6' 

(a) 


d' 
0 

FIG.  97. 


(b) 


two  directions  only,  as  ee* ',  then  a  considerable  part  of  such 
rods  in  the  corners  of  the  square  are  ineffective. 

The  method  of  analysis  of  Art.  150,  Chapter  VI,  may  also 
be  applied  to  this  problem. 

In  the  case  of  cantilever  beams  such  as  in  footings  the 
maximum  shearing  stress  is  near  the  centre  where  the  maxi- 
mum moment  occurs.  Shear  cracks  tend  to  form  on  the  dotted 
curved  lines,  Fig.  97  (b).  Bent  rods,  if  used,  must  be  bent  up 
just  outside  the  column,  and  not  at  the  end  of  the  beam,  arid 
stirrups  must  be  spaced  closely  at  this  point.  The  beam  being 
short  it  may  require  special  attention  to  bond  stress. 

For  large  individual  footings  a  beam  and  slab  may  be  eco- 
nomical. To  secure  the  benefit  of  a  T  section  and  to  give 


332 


BUILDING    CONSTRUCTION. 


[Cn.  VII. 


a  flat  upper  surface  the  beam  may  be  placed  under  the  slab 
as  shown  in  Fig.  98.  This  arrangement  requires  some  atten- 
tion as  to  connection  of  slab  to  beam,  as  the  upward  pres- 
sure against  the  slab  tends  to  pull  it  away  from  the  beam.  The 
use  of  an  extra  horizontal  rod  in  the  top  of  the  main  beam 


FIG.  98. 

bonded  by  stirrups  will  give  a  thoroughly  good  anchorage  for 
the  transverse  rods  of  the  slab.  For  still  larger  areas  a  sys- 
tem of  girders  and  beams  may  be  adopted  constituting  a  floor 
reversed  as  to  loads. 

170.  Walls  and  Partitions. — The  reinforcing  of  these  parts 
is  largely  for  the  purpose  of  preventing  cracks  or  of  localizing 
them  to  desired  lines.  Where  lateral  pressures  occur,  of  course 
the  beam  action  must  be  considered.  Walls  are  usually  3-6 
inches  thick  and  reinforced  both  ways  with  J-  to  J-in.  rods, 
spaced  about  2  feet  apart. 


CHAPTER  VIII. 

ARCHES. 

171.  Advantages  of  the  Reinforced  Arch. — If  the  loads  on 
an  arch  were  all  fixed  loads,  it  would  be  possible  in  any  case 
to  construct  an  arch  ring  so  that  the  resultant  pressure  at 
all  sections  would  intersect  the  centre  of  gravity  of  the  sec- 
tion. The  compressive  stress  at  any  section  would  then  be 
uniformly  distributed  over  the  section,  and  the  arch  would  be 
proportioned  only  for  this  uniform  compression.  The  "line 
of  pressure"  would  lie  at  the  axis  of  the  arch  throughout. 
If,  however,  the  arch  ring  is  not  made  to  fit  the  "  line  of  pres- 
sure", or  if  part  of  the  load  is  a  live  load,  then  the  resultant 
pressure  will  not  in  general  coincide  with  the  axis  of  the  arch. 
There  will  exist  both  bending  and  direct  compression.  If  the 
resultant  pressure  and  its  position  are  known,  the  analysis  of 
the  stresses  at  any  section  is  made  in  accordance  with  the 
method  explained  in  Arts.  80-85,  Chapter  III. 

In  ordinary  masonry  or  concrete  arches  tensile  stresses 
are  not  permissible.  The  ring  must  therefore  be  designed  so 
that  the  line  of  pressure  will  not  pass  outside  the  middle  third. 
In  reinforced  arches  this  limitation  does  not  exist.  The  arch 
rib  is  a  beam,  and  if  properly  reinforced  it  may  carry  heavy 
bending  momoiits  involving  tensile  stresses  in  the  steel. 

Theoretically  the  gain  in  economy  by  the  use  of  steel  in 
a  concrete  arch  is  not  great.  If  the  pressure  line  does  not 
depart  from  the  middle  third,  the  steel  reinforces  only  in  com- 
pression and  in  this  respect  is  not  as  economical  as  concrete. 
If  the  line  of  pressure  deviates  farther  from  the  centre,  result- 
ing in  tensile  stresses  in  the  steel,  the  conditions  are  such  that 

333 


334  ARCHES.  [Cn.  VIII. 

those  stresses  must  be  provided  for  by  the  use  of  the  steel  at 
very  low  working  values.  That  is  to  say,  the  direct  compres- 
sion in  the  arch  is  so  large  a  factor  that  the  limiting  stresses 
in  the  concrete  will  always  result  in  very  small  unit  tensile 
stresses  in  the  steel  where  any  tension  exists  at  all. 

Practically  the  value  of  reinforcement  is  very  considerable. 
It  renders  an  arch  a  much  more  secure  and  reliable  structure, 
it  greatly  aids  in  preventing  cracks  due  to  any  slight  settlement, 
and  by  furnishing  a  form  of  construction  of  greater  reliability 
makes  possible  the  use  of  working  stresses  in  the  concrete 
considerably  higher  than  is  usual  in  plain  masonry.  Further- 
more, in  long-span  arches  where  the  dead  load  constitutes  by 
far  the  larger  part  of  the  load,  any  possible  increase  in  average 
working  stress  counts  greatly  towards  economy.  It  affects 
not  only  the  arch  but  the  abutments  and  foundations. 

172.  Methods  of  Reinforcement. — The  reinforcement  of 
arches  is  arranged  in  various  ways.  Since  the  arch  is  a  beam 
subject  to  either  positive  or  negative  bending  moments  it  is 
essential  that  it  should  be  reinforced  on  both  sides,  but  the 
shearing  stresses  due  to  beam  action  are  relatively  small,  so 
that  little  is  needed  in  the  way  of  web  reinforcement.  The 
arch  is  also  subjected  to  heavy  compression,  so  that  it  is  desir- 
able that  the  inner  and  outer  reinforcement  be  tied  together, 
somewhat  as  in  a  column,  although  in  this  case  the  necessity 
therefor  is  much  less. 

A  large  proportion  of  the  arches  which  have  been  con- 
structed have  been  built  according  to  some  one  of  the  various 
"systems"  that  have  been  devised.  The  most  important  of 
these  systems  are  the  Monier  and  the  Melan.  In  the  Monier 
system,  invented  about  1865,  the  reinforcement  consists  of 
wire  netting,  one  net  being  placed  near  the  intrados  and  one 
near  the  extrados.  The  longitudinal  wires  are  made  smaller 
than  those  following  the  arch  ring,  as  they  serve  only  to  aid 
in  equalizing  the  load  and  in  preventing  cracks.  A  large  num- 
ber of  bridges  have  been  built  in  Europe  on  this  system. 

In  the  Melan  type,  invented  about  1890,  the  steel  is  in 


§  173.]  GENERAL  METHOD  OF  PROCEDURE.  335 

the  form  of  ribs  of  rolled  I  sections,  or  of  built-up  lattice  gir- 
ders, which  are  spaced  two  to  three  feet  apart.  The  flanges 
constitute  the  principal  reinforcement,  but  the  web  enables 
the  steel  frame  to  be  self-supporting  and  to  carry  shearing 
stresses,  and  in  the  open  lattice  type  it  furnishes  a  good  bond 
with  the  concrete.  The  Melan  arch  has  been  built  extensively 
in  this  country,  largely  under  the  direction  of  Mr.  Edwin 
Thacher. 

Many  arches  are  now  being  constructed  in  which  reinforcing 
bars  of  any  satisfactory  form  are  employed  without  reference 
to  any  particular  system,  being  used  in  accordance  with  the 
requirements  of  the  case.  The  problem  of  reinforcement  is 
quite  as  simple  as  in  a  beam,  after  the  moments  and  thrusts 
in  the  arch  have  been  found. 

ANALYSIS   OF  THE   ARCH. 

173.  General  Method  of  Procedure. — The  method  of  analysis 
presented  here  is  based  on  the  elastic  theory  and  is  of  gen- 
eral application  to  arches  of  variable  moment  of  inertia  and 
loaded  in  any  manner.  It  is  mainly  an  algebraic  method, 
although  certain  simple  graphical  aids  may  be  used  advan- 
tageously. It  necessarily  assumes  that  a  preliminary  design 
has  been  made  by  the  aid  of  approximate  or  empirical  rules 
or  by  reference  to  the  proportions  of  existing  arches.  This 
arch  is  then  exactly  analyzed  and  the  results  used  in  cor- 
recting the  design;  the  corrected  design  may  then  in  turn  be 
analyzed  if  it  departs  too  greatly  from  the  one  first  assumed. 
A  discussion  of  the  various  rules  for  thickness  of  crown  and 
form  of  arch  will  not  be  entered  upon  here.  For  this  infor- 
mation the  reader  is  referred  to  the  various  treatises  on  the 
arch,  and  especially  to  those  of  Professor  Cain  and  Professor  M. 
A.  Howe.  The  work  of  Professor  Howe  on  "  Symmetrical 
Masonry  Arches"*  contains  a  very  useful  table  of  data  of 
existing  masonry  and  reinforced-concrete  arches. 

*  New  York,  1906. 


336  ARCHES.  [Cn.  VIII. 

The  analysis  of  an  arch  consists  in  the  determination  of 
the  forces  acting  at  any  section,  usually  expressed  as  the 
thrustj  the  shear  and  the  bending  moment,  at  such  section. 
The  thrust  is  here  taken  to  be  the  component  of  the  resultant 
parallel  to  the  arch  axis  at  the  given  point,  and  the  shear  is 
the  component  at  right  angles  to  such  axis.  The  thrust  causes 
simple  compressive  stresses;  the  shear  causes  stresses  similar 
to  those  produced  by  the  vertical  shear  in  a  simple  beam. 

The  method  of  procedure  will  be  to  determine,  first,  the 
thrust,  shear,  and  bending  moment  at  the  crown.  These  being 
known,  the  values  of  similar  quantities  for  any  other  section 
can  readily  be  determined.  A  length  of  arch  of  one  unit  will 
be  considered. 

174.  Thrust,  Shear,  and  Moment  at  the  Crown  (H0,Vo,M0). 

Notation.  (See  Fig.  100.) 

Let  HQ  =  thrust  at  the  crown; 
V0  =  shear  at  the  crown; 

MQ  =  bending  moment  at  the  crown ?  assumed  as 
positive   when    causing    compression    in 
the  upper  fibres; 
N,  V,  and  M  =  thrust,  shear,  and    moment  at  any  other 

section; 
R  =  resultant  pressure  at  any  section  =  resultant 

of  N  and  7; 

ds  =  length  of  a  division  of  the  arch  ring  meas- 
ured along  the  arch  axis; 
n  — number  of  divisions  in  one-half  of  the  arch; 
/  =  moment  of  inertia  of  any  section  =  7concrete 

+  wasted  (seep.  92); 
P=any  load  on  the  arch; 

x,  y  =  co-ordinates  of  any  point  on  the  arch  axis 
referred  to  the  crown  as  origin,  and  all 
to  be  considered  as  positive  in  sign; 
m  =  bending  moment  at  any  point  in  the  canti- 
lever, Fig.  100,  due  to  external  loads. 


§  174.]     THRUST,  SHEAR,  AND  MOMENT  AT  THE  CROWN.       337 

Let  AB,  Fig.  99,  be  a  symmetrical  arch  loaded  in  any  man- 
ner with  loads  P1}  P2,  etc.  Divide  the  arch  into  an  even  num- 
ber of  divisions  (ten  to  twenty  usually),  making  the  divisions 
of  such  a  length  that  the  ratio  ds:I  will  be  constant.  This 
may  be  done  by  trial  or  by  the  more  direct  method  explained 
in  Art.  178.  Mark  the  centre  point  of  each  division  and  num- 


FIG.  99. 


ber  the  points  as  shown.  Consider  the  arch  to  be  cut  at  the 
crown  and  each  half  to  act  as  a  cantilever  subjected  to  exactly 
the  same  forces  as  exist  in  the  arch  itself,  that  is,  the  given 
loads,  the  reactions,  and  the  stresses  at  the  crown,  represented 


FIG.  100. 

by  H0)  Fo,  and  M0  (Fig.  100).    H0  is  applied  at  the  gravity 
axis. 

Let  m= bending  moment  at  any  point,  1,  2,  3,  etc.,  due  to 
the  given  external  loads  P  (considering  the  arch  as  two  canti- 
levers). Denote  by  mR  the  moments  in  the  right  half  and 
by  mL  those  in  the  left  half  of  the  arch.  All  the  moments  m 


338  ARCHES.  [Cn.  VIII. 

^ 

will  be  negative.    The  values  of  H0,  VQ,  and  MQ  will  then 
be  given  by  the  following  equations: 

„.      nlmy—  2m  Zy 

'     '     '     ' 


In  these  equations  the  summations  Zy,  Iy2,  and  Ix2  are 
for  one-half  of  the  arch  only;  the  summation  Im  is  for  the 
entire  arch  and  is  equal  to  2mR+  ZmL;  the  summation 
2(niR—mL)x  is  a  summation  of  the  products  (mR—m^x,  in 
which  ra#  and  m^  are  the  bending  moments  at  corresponding 
points  in  the  right  and  left  halves  which  have  equal  abscissas  x\ 
and  the  summation  2my  is  for  the  entire  arch,  but  since  sym- 
metrical points  have  equal  y's  this  quantity  may  be  calculated 
as  2(mR+mL)y.  A  positive  result  for  70  indicates  action  as 
shown  in  Fig.  100.  All  quantities  are  readily  calculated 
Distances  should  be  scaled  and  quantities  tabulated.* 

*  Demonstration.  —  Consider  the  left-hand  cantilever  of  Fig.  100.  Under 
the  forces  acting  the  point  C  will  deflect  and  the  tangent  to  the  axis  at  this 
point  will  change  direction  (the  abutment  at  A  being  fixed).  Let  Ayt  Ax, 
and  4$  be,  respectively,  the  vertical  and  horizontal  components  of  this 
motion  and  the  change  in  angle  of  the  tangent.  Then  according  to  the  prin- 
ciples relating  to  curved  beams  l  we  have  the  values 


Jy=IMx~      Ax-IMy,    and      A^=IM~t   ...     (a) 

in  which  the  various  quantities  have  the  same  significance  as  in  Art.  174. 

In  like  manner,  referring  to  the  right  cantilever,  let  Ay'  ,  Ax'  ,  and  A<j>' 
represent  the  components  of  the  movement  of  C  and  the  change  of  angle  of 
the  tangent.  These  may  be  expressed  in  terms  similar  to  Eq.  (a). 

Now  evidently 

Ay=Ay't     Ax=-Ax't     and     A$=  -  A$  .....     (6) 

Furthermore,  since  ds/I  is  constant  and  likewise  E,  the  quantity  ds/EI  may 
be  placed  outside  the  summation  sign. 

1  See  Church's  Mechanics,  or  Johnson's  Framed  Structures,  p.  236. 


§  176.]  PARTIAL  GRAPHICAL  CALCULATION.  339 

175.  Thrust,  Shear,  and  Moment  at  any  Section. — The 

values  of  H0,  VQ,  and  M0  having  been  found,  the  total  bend- 
ing moment  at  any  section,  1,  2,  etc.,  is 

M=m+MQ+H0y±Vox.       ....    (4) 

The  plus  sign  is  to  be  used  for  the  left  half  and  the  minus  sign 
for  the  right  half  of  the  arch. 

The  resultant  pressure,  R,  at  any  section  of  the  arch  is 
equal  in  magnitude  to  the  combined  resultant  of  the  external 
loads  between  the  crown  and  the  section  in  question,  and  the 
forces  HQ  and  VQ.  These  resultants  can  best  be  found  graphic- 
ally. The  thrust,  N,  is  the  component  of  the  resultant,  Rt 
parallel  to  the  arch  axis  and  the  shear,  V,  is  the  component 
perpendicular  to  this  axis. 

176.  Partial  Graphical  Calculation.— Where  the  loads  are 
vertical  the  calculation  of  the  quantities  m  can  be  advan- 
tageously performed  by  means  of  an  equilibrium  polygon  as 
follows : 

Let  AB,  Fig.  101,  represent  the  arch  axis.  The  load  line  is 
a-c-b.  Select  any  convenient  pole  0  on  a  horizontal  line  through 

Using  the  subscript  L  to  denote  left  side  and  R  to  denote  right  side  we 
then  derive  the  relations 

ZMRx,  -I 
-IMRy,  I .     .      (c) 

IML  =  -  ZMR.   J 

The  moment  M  may  in  general  be  expressed  in  terms  of  known  and  un- 
known quantities  thus: 

ML=VIL+ M0+ H0y+  V^x  for  the  left  side 
and 

Mft--=mR+  M0+H0y  —  VoX  for  the  right  side. 

Hence,  substituting  in  (c)  and  combining  terms,  and   noting  that  2M9  for 
one  half  is  equal  to  nM 0,  we  have 


.     ......    (d) 

y*  =  0,  .     .     .     .     .      (e) 
+2nM0+  2H02y=Q,       .     .     .     .     .      (/) 

From  Eq.  (d)  is  obtained  Eq.  (2),  p.  268;  and  from  Eqs.  (e)  and  (/)  art 
obtained  Eqs.  (1)  and  (3),  noting  that  Zmi,+  ImR  =  Zm,  and  Imi,y+  2mRy= 
Imy. 


340 


ARCHES. 


[CH.  VTH. 


the  point  c,  at  the  junction  of  loads  P3  and  P4,  the  loads  adja- 
cent to  the  crown  C.  Construct  the  equilibrium  polygon  efgh, 
producing  to  i  and  k  the  segment  fg  between  loads  P3  and  P4. 
Drop  verticals  from  the  points  1,  2,  3,  etc.  The  desired  bend- 
ing moments  m,  at  the  several  points,  will  then  be  equal  to 
the  corresponding  intercepts  z2,  £3,  etc.,  on  these  verticals 
between  the  equilibrium  polygon  and  the  line  ik,  multiplied 
by  the  pole  distance  H\  or  in  general  m=Hz. 


FIG.  101. 

Finally,  after  the  values  of  H0,  70,  and  M0  are  found  by 
Eqs.  (1),  (2),  and  (3),  the  true  equilibrium  polygon  can  be 
drawn,  if  desired,  and  values  of  thrust,  shear,  and  moment 
at  various  points  determined  graphically.  The  true  pole  is 
found  by  laying  off  V0  and  H0  from  the  point  c.  The  dis- 
tance of  the  equilibrium  polygon  above  or  below  the  axis  at 
the  crown  is  equal  to  MQ/H0.  It  lies  above  the  axis  if  the 
result  is  positive  and  below  if  negative.  The  equilibrium 
polygon  is  then  drawn  from  the  crown  each  way  towards  the 
ends. 


§  178.J  DIVISION  OF  ARCH  RING.  341 

Where  the  loads  are  inclined  at  various  angles  it  is  still 
possible  to  use  a  graphical  process  for  getting  values  of  m, 
but  there  is  little  to  be  gained  in  such  a  case.  After  the  values 
of  #o,  VQ>  and  MQ  are  found,  however,  it  will  be  expedient  to 
draw  a  final  equilibrium  polygon,  or  line  of  pressure,  as  ex- 
plained above. 

177.  General  Observations. — The  method  of  analysis  just 
described  is  brief,  general,  and  easily  followed.  The  arithmetical 
calculations  are  not  longer  than  those  required  in  the  usual 
graphical  process,  while  the  graphical  aids  here  suggested  are 
of  the  simplest  character. 

The  loads  and  their  points  of  application  have  been  con- 
sidered apart  from  the  divisions  of  the  arch  ring,  as  the  two 
things  are  in  no  wise  related.  Where  no  spandrel  arches  are 
used  and  the  entire  load  is  applied  continuously  along  the 
arch  ring,  the  load  may  for  convenience  be  divided  to  corre- 
spond with  the  arch  divisions  and  applied  at  the  center  points, 
1,  2,  3,  etc.  This  division  is,  however,  of  no  importance,  the 
only  requirement  being  a  sufficiently  small  subdivision  of  the 
arch  ring  and  of  the  load  so  that  the  errors  of  approximation 
will  be  negligible.  Where  spandrel  arches  are  used,  the  live 
load  and  a  large  part  of  the  dead  load  will  be  applied  at  the 
centers  of  the  arch  piers.  The  weight  of  the  main  arch  ring 
may  also  be  considered  as  concentrated  at  these  same  points. 

If  calculations  are  to  be  made  for  more 'than  one  loading 
it  will  be  noted  that  the  denominators  of  the  values  for  HQ, 
V0,  and  M0  do  not  change.  The  quantities  involving  ra  are 
the  only  ones  requiring  recalculation,  and  if  the  load  on  but 
one-half  of  the  arch  is  changed,  then  the  values  of  m  for  that 
half  only  need  be  recalculated.  In  the  case  of  a  symmetrical 
loading,  or  a  load  on  one-half  only,  the  calculation  of  m  is 
also  necessary  for  one-half  the  arch  only.  For  symmetrical 
loads,  T70  =  0. 

178.  Division  of  Arch  Ring  to  give  Constant  ds/I. — In  most 
cases  the  depth  of  the  arch  ring  increases  from  crown  towards 
springing  line,  giving  a  variable  moment  of  inertia.  Consider- 


342  ARCHES.  [Cn.  Vill. 

ing  the  concrete  only,  the  moment  of'  inertia  will  increase  as  d3 
so  that  a  comparatively  small  change  in  depth  will  cause  a 
large  change  in  moment  of  inertia.  To  maintain  ds/I  con- 
stant, the  value  of  ds  will  therefore  be  much  greater  near  the 
springing  line  than  at  the  crown,  and  hence  to  secure  the  de- 
sired accuracy  the  length  of  division  at  the  crown  will  need 
to  be  made  fairly  short.  The  value  of  ds/I  to  adopt  so  that 
there  will  be  no  fractional  division  may  be  determined  as  fol- 
lows: 

Let   i= 


value  of  i; 
s  =  half    length   of   the  arch  ring  measured  along  the 

axis; 
n  =  desired  number  of  divisions  in  one-half  the  arch. 

Calculate  first  the  mean  value  of  i  for  the  half  arch  ring  by 
determining  several  values  at  equal  intervals  along  the  arch. 
Then  the  desired  value  of  ds/I  is 


s 


The  value  of  ds/I  being  known,  the  proper  length  of  ds  for 
any  part  of  the  arch  ring  can  readily  be  determined.  Begin- 
ning at  the  crown,  the  length  of  the  first  division  is  deter- 
mined, then  the  second,  third,  etc.,  to  the  end.  The  length 
of  a  division  not  being  exactly  known  beforehand,  the  value 
of  /  for  that  division  will  not  be  exactly  known,  but  the  neces- 
sary adjustment  is  very  simple. 

In  determining  the  value  of  7  the  steel  reinforcement  must 
be  duly  considered. 

179.  Temperature  Stresses. — For  a  rise  of  temperature  of  t 
degrees  the  increase  in  span-length,  if  the  arch  be  not  re- 
strained, would  be  equal  to  ctl,  where  c  =  coefficient  of  expan- 


§  180.]         STRESSES  DUE   TO  SHORTENING  OF  ARCH.  343 

sion  and  Z  =  span.     The  restraint  of  the  abutments  induces  a 
thrust  #o  at  the  crown  given  by  the  equation 

=  EI_  ctln          * 

-'*2   .....     • 


The  summations  refer  to  one-half  the  arch. 
Having  determined  H0,  then 


(7) 


The  bending  moment  at  any  point  is 

(8) 


The  thrust  and  shear  at  any  point  in  the  arch  are  found 
by  resolving  H0  parallel  and  normal  to  the  arch  axis  at  that 
point.     Graphically,  the  true  equilibrium  polygon  is  a  hori- 
}      zontal  line  drawn  a  distance  below  the  crown  equal  to  M0/H0 
=  Iy/n. 

180.  Stresses  Due  to  Shortening  of  Arch  from  Thrust.—  A 
thrust  throughout  the  arch  producing  an  average  stress  on 
the  concrete  equal  to  fc  lbs/in2  would  shorten  the  arch  span 
an  amount  equal  to  fcl/E  if  unrestrained.  This  action  develops 
horizontal  reactions  in  the  same  manner  as.  a  lowering  of  tem- 

*  Demonstration.  —  For  temperature   stresses   J<£  of  Eq.    (a),  p.   338,  is 
zero  and  Ax  is  equal  to  the  change  in  length  of  the  half-span,  =C~.     We 

therefore  have 

„,,      ds     ctl 
IMLy  —=—, 

and 


In  this  case,  there  being  no  external  loads,  m=0,  and  from  symmetry, 
F0=0,  hence  M  =  MQ+H0y.  Substituting  this  value  of  M  in  the  above 
equations  we  have 

Ctl      El 


and 

n 

From  these  are  readily  derived  Eqs.  (6)  and  (7). 


344  ARCHES.  [On.  VIII. 

perature.  The  value  of  the  resulting  reactions,  or  the  crown 
thrust,  may  then  be  found  by  substituting  fel/E  for  ctl  of 
Eq.  (6).  There  results 

I_  fcln 

'' 


The  moments  at  crown  and  elsewhere  are  given  by  Eqs.  (7) 
and  (8),  using  the  value  of  H0  from  Eq.  (9). 

The  thrusts  and  moments  due  to  arch  shortening  will  not 
usually  be  large.  They  may  be  applied  as  corrections  to  the 
thrusts  and  moments  found  before. 

181.  Deflection  of  the  Crown.  —  The  downward  deflection 
of  the  crown  under  a  load  is  given  by  Eq.  (a),  p.  268.  It  is 


(10) 


If  M  is.  not  determined  for  all  points,  use  the  value  of  M 
from  Eq.  (4),  deriving 


.     (11) 


The  summations  are  for  one-half  only. 

The  rise  of  crown  due  to  an  increase  of  temperature  is  ob- 
tained from  Eq.  (11)  by  substituting  from  Eqs.  (6)  and  (7). 
There  results 

,;..   ctl  nlxy-ZxZy^ 


182.  Unsymmetrical  Arches.  —  If  the  arch  is  unsymmet- 
rical  the  value  of  ds/I  should  be  made  constant  for  the  entire 
arch,  and  the  number  of  divisions  made  even  as  before.  The 
central  point  of  the  arch  with  reference  to  the  number  of  divi- 
sions may  then  be  taken  as  the  crown,  and  the  X-axis  made 
tangent  to  the  arch  at  this  point.  The  two  halves  of  the  arch 
are  now  unlike  and  all  terms  resulting  from  substitution  in 
Eq.  (c),  p.  269,  must  be  retained.  Explicit  formulas  for  H0, 


§  183.]         APPLICATION  OF  THE  PRECEDING  THEORY. 


345 


Fo,   and  MQ  are  very  cumbersome,  but  the  three  equations 

derived  from  (c)  are  as  follows: 

(ILx-  IRx)M0+  (2Lxy-  2Rxy)H0+  Ix2V0  =  IRmx-  ZLmx,  (13) 

.  .  (14) 
.  .  (15) 

Where  no  subscript  is  used  the  summation  is  for  the  entire 
arch.  Numerical  values  of  the  coefficients  of  M  0,  HQ,  and  FO 
should  be  obtained  and  the  equations  then  solved. 
JiH^Appfication  of  the  Preceding  Theory.  —  Example  1.  — 
An  arch  ring  will  be  assumed  of  the  dimensions  shown  in 
Fig.  102.  Span  length  with  reference  to  the  axis  =  30  ft., 
rise  =  8  ft.  Thickness  at  crown  =  1  ft.,  at  springing  lines  = 
1  ft.  6  in.  For  a  small  arch  such  as  this  great  accuracy  is  not 
needed,  hence  a  small  number  of  divisions  may  be  used.  The 
number  will  be  four  for  each  half.  These  divisions  are  deter- 
mined so  that  ds/I  is  constant.  The  loads  are  applied  at 
the  centre-points  1,  2,  3,  4,  and  are  assumed  to  be  somewhat 
inclined  (excepting  loads  P4  and  P5),  the  several  vertical  and 
horizontal  components  being  as  given  in  the  figure. 


TABLE  A. 
CALCULATIONS  FOR  H0, 


AND  M. 


1 

o 

3 

4 

5 

6 

7 

8 

9 

Point. 

X 

y 

X* 

y* 

mL 

MR 

(mL+mR)y 

(mR-mL)x 

1 
2 
3 
4 

1.55 
4.90 
8.45 

12.85 

.09 
.68 
2.10 
5.35 

2.40 
24.01 
71.40 
165.12 

.01 

.46 
4.41 

28.62 

0 
-   13,840 
-  46,800 
-116,680 

0 
-     8,640 
-  29,560 
-  75,490 

0 
-  15,300 
-  160,400 
-1,028,100 

0 
+  25,500 
+  145,700 
+  529,300 

I 

15.00 

S.22 

262.93 

33.50 

-  1  ,203,800 

+  700,500 

-291,010 

Spring- 
ing. 

8.00 

-180,820 

-120,640 

4(-  1203300)  -(-291010X8.22) 
2[(8.22)2-4X33.5Q]  - 

°+1'330  lbs' 


=  + 18,230  Ibs. 


»•/,%• 

Eq.  (1)  gives 

Eq.  (2)  gives 

-291010+2X18230X8.22 
Eq.  (3)  gives  M0=  --        --    =  -1,090  ft-lbs. 


346 


ARCHES. 


[Cn.  Vin. 


TABLE  B. 

BENDING  MOMENTS,  THRUSTS,  AND  ECCENTRIC  DISTANCES. 


1 

2 

3 

4 

5 

6 

7 

9 

9 

Bending  Moment  M  . 

Thrusts. 

Eccentric 
Distances. 

Point. 

H0y 

Vox 

Left. 

Right. 

Left. 

Right. 

Left. 

Right. 

1 

1,640 

2,060 

+  2,610 

-1,520 

18,450 

18,640 

+  .14 

-.08 

2 

12,400 

6,530 

+  4,000 

-3,860 

19,580 

19,310 

+  .21 

-.20 

3 

38,300 

11,260 

+  1,650 

-3,620 

22,050 

20,770 

+  .07 

-.17 

4 

97,500 

17,120 

-3,100 

+  3,850 

28,800 

24,970 

-.11 

+  .15 

Spring- 
ing. 

145,900 

20,000 

-16,070 

+  4,140 

28,800 

24,970 

-.56 

+  .17 

The  calculations  of  the  several  quantities  in  the  formulas 
for  //o,  T70,  and  M0  (p.  338)  are  given  in  Table  A.  The  co- 
ordinates x,  y  of  the  several  points  are  given  in  cols.  2  and 
3;  then  x2  and  y2  in  cols.  4  and  5;  then  in  cols.  6  and  7  are 
given  the  quantities  mL  and  mR,  considering  each  half -arch  a 
cantilever.  These  are  readily  calculated.  Thus,  on  the  left, 
for  point  1,  ra=0;  for  point  2,  m=  4130  X  (4.90 -1.55)  =  13, 840; 
for  point  3,  m =4130  X  (8.45  - 1 .55)  +  5035  X  (8.45  -  4.90)  +  310  X 
(2.10 -.68)  =46,800;  and  for  point  4,  m  =  4130X  (12.85-1.55)  + 
5035  X  (12.85  -  4.90)  +  5950  X  (12.85  -  8.45)  +  310  X  (5 .35  -  .68)  + 
725(5.35 -2.10)  =116,680.  The  value  of  m  at  the  springing 
line  is  also  calculated  and  placed  in  this  table  for  future  use. 
The  moments  on  the  right  are  similarly  found .  All  moments  m 
are  negative.  In  cols.  8  and  9  are  then  given  the  products 
(mL + mR)y  and  (niR — mL)x. 

Substituting  in  Eqs.  (1),  (2),  and  (3),-  p.  268,  there  are  ob- 
tained the  values  for  HQ,  T'o,  and  MO  given  below  the  table. 

The  values  of  the  .bending  moments,  thrusts,  and  shears  at 
any  point  may  now  be  found  either  graphically  or  algebraically. 
The  force-diagram  method  will  be  much  the  better  for  obtaining 
thrusts  and  shears;  the  moments  may  then  be  obtained  either 


§  183.]        APPLICATION  OF  THE   PRECEDING   THEORY. 


347 


by  constructing  an  equilibrium  polygon  or  by  the  application 
of  Eq.  (4),  p.  339. 

In  Fig.  102  the  graphical  construction  is  given.  The  load- 
line  is  a-c-b.  The  true  pole  is  found  by  laying  off  70=  +  1330 
from  point  c  (at  the  junction  of  the  loads  adjacent  to  the 
-crown,  P4andP5);  then  #0  =  18,230  horizontally  to  0.  The 


P,   e* 

P5 

^s=^ 

MH 

^T; 

"  i 

/ 

a  «     i, 

ii 

\: 

! 

1600 


force  diagram  is  drawn  and  then  the  equilibrium  polygon, 
beginning  at  the  crown  and  drawing  the  segment  1  —  1  at  a 
distance  below  the  crown  equal  to  1090/18230  =  .06  ft.  The 
resultant,  R,  acting  at  any  section  may  be  scaled  from  the 
force  polygon,  and  the  moment  at  any  point  will  be  equal  to 
this  resultant  multiplied  by  the  perpendicular  distance  at. that 
point  from  the  arch  axis  to  the  equilibrium  polygon.  For  ex- 
ample, the  bending  moment  at  point  g  is  equal  to  the  force  Od 
multiplied  by  the  arm  gk.  The  tangential  component  of  the 


348  ARCHES.  [Cn.  VIII. 

resultant  R  (the  true  thrust  A7)  may  be  found  by  resolving  the 
force  R  parallel  and  normal  to  the  arch  axis  at  the  point  in 
question.  In  most  cases  the  thrust  AT  may  be  taken  as  equal 
to  R.  The  shear  V  will  be  the  normal  component  of  R'}  it  will 
not  usually  require  consideration. 

Table  B  contains  calculated  values  of  moments  and  eccen- 
tric distances  for  points  1,  2,  3,  4  and  the  springing  lines.  The 
moments  are  calculated  from  the  formula  (Eq.  (4))  M- 
m-}~Mo  +  Hoy±VQX.  The  quantities  m  are  obtained  from 
Table  A,  cols.  6  and  7.  The  thrusts  are  scaled  from  the  force 
polygon,  being  in  each  case  the  thrust  on  the  abutment  side  of 
the  point  in  question.  The  eccentric  distances  are  equal  to 
the  moments  divided  by  the  thrusts;  they  are  of  use  in  cal- 
culating stresses  in  the  arch.  Obviously  the  bending  moment 
at  any  other  point,  such  as  g,  may  be  calculated  in  the  same 
Avay  as  those  here  given. 

184.  Example  2.  (Fig.  103.) — For  another  example  an  arch 
will  be  assumed  of  100-ft.  span  and  20-ft.  rise;  thickness  at 
crown  =  30  in.,  thickness  at  springing  line  =42  in.  It  will  be 
assumed  that  the  roadway  is  supported  on  spandrel  piers  10  ft. 
apart,  thus  concentrating  most  of  the  load  at  points  10  ft. 
apart  as  shown;  the  weight  of  the  arch  ring  will  also  be  as- 
sumed as  applied  at  these  points.  The  loads  as  given  represent 
an  arch  with  live  load  on  the  left  half  only.  The  half  arch  is 
divided  into  ten  divisions,  making  ds/I  constant.  The  loads  in 
this  case  are  vertical,  so  that  the  graphical  method  may  be  used 
to  advantage  in  determining  the  cantilever  moments  m.  The 
load  line  a-c-b  is  drawn  and  a  pole  0'  selected  on  a  horizontal 
line  through  c  at  the  center  of  the  crown  load  P5.  The  pole 
distance  is  H.  An  equilibrium  polygon,  efgh,  is  then  drawn, 
and  the  moments  m  will  be  equal  to  the  intercepts,  z,  from  this 
polygon  to  the  horizontal  line  ik,  multiplied  by  H.  These 
moments,  and  the  remainder  of  the  calculations  for  HQ,  VQ,  and 
MO  are  given  in  Table  C.  The  true  pole  0  is  then  found  as 
before  and  the  correct  equilibrium  polygon  drawn.  The 
thrusts  are  then  scaled  from  the  force  polygon,  and  the  eccentric 


§  184.]         APPLICATION  OF  THE  PRECEDING  THEORY.  349 


350 


ARCHES. 


[CH.  VIII. 


distances  from  the  equilibrium  polygon.  These  are  given  in 
Table  D,  together  with  resulting  bending  moments.  The  bend- 
ing moments  may  also  be  calculated  as  done  in  Example  1. 

TABLE  C. 
CALCULATIONS  FOR  H0,  F0,  AND  AT0. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

Point 

X 

y 

a,2 

2/2 

mL 

mR 

(mL  +  mR)u 

(mK-mL)x 

1 

2.07 

.07 

4.3 

.00 

-       13,500 

-       13,500 

2,000 

2 

5.96 

.30 

35.5 

.09 

-       38,700 

-      38,700 

23,000 



3 

10.00 

.72 

100.0 

.52 

-       65,000 

-      65,000 

94,000 

4 

14.18 

1.40 

201.1 

1.96 

-     148,600 

-     127,700 

-       387,000 

+       296,000 

5 

18.43 

2.37 

339.7 

5.62 

-     233,600 

-     191,400 

-    1,007,000 

+      777,000 

6 

23.06 

3.77 

531.8 

14.21 

-     369,000 

-     288,400 

-    2,479,000 

+    1,859000 

I 

28.06 
33.60 

5.64 
8.23 

787.4 
1129.0 

31.81 
67.73 

-     539,000 
-    781,400 

-     408,400 
-     577,400 

-    5,348,000 
-11,183,000 

+    3,665,000 
+    6,854,000 

9 

39.60 

11.80 

1568.2 

139.24 

-1,075,400 

—     781,400 

-21,910,000 

+  11,642.000 

10 

46.  40 

16.60 

2153.0 

282.24 

-1,511,000 

-1,083,000 

-43,580,000 

+  19,859,000 

2 

51  .  10 

6850.0 

543.42 

-86,013,000 

+  44,952,000 

-8,350,100 

u o=  _-8350100  +  2XTI>760X61.10 

TABLE  D. 

THRUSTS,  ECCENTRIC  DISTANCES,  AND  MOMENTS. 


1 

2 

3 

4 

5 

6 

7 

Thrusts. 

Eccentric  Distances. 

Bending  Moments. 

Point. 

Left. 

Right 

Left. 

Right. 

Left, 

Right. 

1 

76,800 

77,400 

+  .31 

+  .13 

+  23,700 

+    10,100 

2 

76,800 

77,400 

+  .38 

-.13 

+  29,200 

-  10,100 

3 

78,600 

78,800 

+  .61 

-.22 

+  48,000 

-   17,300 

4 

78,600 

78,800 

+  .39 

-.53 

+  30,700 

-  41,700 

5 

78,600 

78,800 

+  .44 

-.57 

+  34,700 

-   45000 

6 

82,700 

81,500 

+  .26 

-.61 

+  21,200 

-  49,700 

7 

82,700 

81,500 

+  .14 

-.52 

+  11  400 

-  42,400 

8 

89,300 

85,300 

-.15 

-.36 

-13,400 

-  30,700 

9 

89,300 

85,300 

-.16 

+  .23 

-14,300 

+    19,600 

10 

Springing 

98,600 
98,600 

90,700 
90,700 

-.44 
-.21 

+  .88 
+  1.67 

-43,400 
-20,700 

+    80,000 
+  152,000 

§  185  ]  MAXIMUM  STRESSES.  351 

Temperature  Stresses.  —  Suppose  in  Ex.  2  it  is  desired  to 
know  the  thrust  and  bending  moment  at  the  crown  due  to  a 
rise  of  temperature  of  30°.  Eqs.  (6)  and  (7),  Art.  179,  will  be 
used.  Assume  E  =  2,000,000  Ibs  /in2  =  288,000,000  Ibs/f  t2.  Sup- 
pose the  value  ds/I,  in  foot-units,  is  3.1.  Then  from  Eq.  (6) 

288,000,000     .000006X30X100X10  _          . 
UQ~        3.1          '   2(10  X  543  -(51.1)2)   =  >S>' 

M0  =  -  2970  X       -  =  -  15,200  f  t-lbs. 


The  equilibrium  polygon  is  a  horizontal  line  drawn  a  distance 
below  the  crown  equal  to  15200/2970-5.11  ft.  The  moment 
at  any  point  is  equal  to  the  thrust  HQ  multiplied  by  the  vertical 
distance  from  such  point  to  this  equilibrium  polygon.  At  the 
springing  line,  M  =  H<*X  (20-5.11)  =2970X14.89=44,200  ft- 
Ibs.  This  may  also  be  calculated  by  Eq.  (8). 

Stresses  Due  to  Shortening  of  Arch.  —  The  modification  of 
the  thrust  due  to  the  compressive  deformations  of  the  arch 
ring  is  found  by  Eq.  (9).  The  average  compressive  stress  at 
any  section  is  found  by  dividing  the  thrust  at  that  section  by 
the  area  of  the  transformed  section  of  arch  ring.  This  is  nearly 
uniform  throughout  the  arch  and  equal  to  about  150  lbs/in2. 
Then. 

1      150X144X100X10  ]b 

~3lX2[10X543-(51.1)2]  " 

This  thrust  is  equal  to  42%  of  the  thrust  due  to  temperature 
change,  already  found.  The  resulting  moments  and  stresses 
will  then  be  42%  of  those  due  to  temperature  change.  They 
will  be  of  opposite  sign. 

185.  Maximum  Stresses  in  the  Arch  Ring.—  From  the 
values  of  thrust,  moment,  and  eccentric  distance,  as  given  in 
Tables  B  and  D,  the  stresses  in  the  concrete  and  steel  can  be 
found  at  any  section  of  the  arch,  as  explained  in  Chap.  Ill  and 
also  in  Art.  147,  Chap.  VI.  The  maximum  value  of  fibre  stress 


352  ARCHES  [Cn  VIII. 

will  be  where  the  sum  of  the  stresses  due  to  thrust,  N,  and 
moment,  M,  is  a  maximum.  This  will  not  in  general  be  where 
either  the  thrust  or  the  moment  is  a  maximum;  but  as  the 
thrust  varies  slowly  along  the  arch  ring  the  maximum  stress  will 
occur  very  near  to  the  point  of  maximum  moment. 

The  position  of  live  load  causing  maximum  moment  at  any 
point  will  differ  in  arches  of  different  proportions.  In  designing 
an  arch  it  is  sufficient  generally  to  determine  the  maximum 
stresses  at  the  crown,  the  haunch,  and  the  springing  line.  This 
will  require  several  different  positions  of  the  live  load.  For  the 
crown  the  maximum  positive  moments  are  caused  when  a 
short  length  of  the  arch  (one-fourth  to  one-third)  at  the  center 
is  loaded,  and  the  maximum  negative  moments  when  the  re- 
maining portions  are  loaded.  The  maximum  positive  and 
negative  moments  at  the  haunch  (about  the  J  point)  are  caused 
when  the  arch  is  loaded  about  two-thirds  the  span  length  and 
one-third  the  span  length  respectively.  The  same  loading  \\  ill 
give  practically  the  maximum  moments  at  the  springing  lines. 

These  conditions  make  it  desirable  to  analyze  the  arch  for 
various  assumed  loadings  about  as  follows:  full  load;  one-third 
of  span  loaded;  two- thirds  of  span  loaded;  center  third  loaded; 
and  end  thirds  loaded.  In  the  case  of  large  and  important 
structures  it  may  be  found  desirable  to  place  the  loads  some- 
what differently  than  here  indicated.  A  complete  and  exact 
solution  can  readily  be  made  by  analyzing  the  arch  for  a  load 
of  unity  at  each  load-point  of  one-half  of  the  arch.  Influence 
lines  can  then  be  drawn  for  moment  or  fibre  stress  and  the 
exact  maximum  values  determined. 

185^.  Example  of  Complete  Analysis  for  Maximum 
Stresses. — To  further  illustrate  the  methods  of  analysis  here 
described,  and  the  use  of  influence  lines  in  determining  max- 
imum stresses,  a  complete  analysis  will  be  made  of  the  arch  of 
Fig.  103,  modified,  for  sake  of  illustration,  by  the  addition  of 
steel  reinforcement  amounting  to  2  in2  per  foot  of  arch  along 
both  the  extrados  and  the  intrados,  and  placed  3  in.  from  the 
surface. 


§  I860.] 


MAXIMUM    STRESSES. 


353 


Calculation  of  Values  of  I  and  of  ds. — The  half  length  of  the 
arch  axis  is  found  to  be  55.17  ft.  The  depth  at  crown  =  2.5  ft., 
and  at  springing  line  =  3. 5  ft.  The  moment  of  inertia  at  any 
section  =  /=/c  +  15L,  where  Ic  and  /«  =  moment  of  inertia  of 
the  concrete  and  steel  sections  respectively.  Following  the 
procedure  of  Art.  178,  the  half  arch  will  first  be  divided  into 
a  convenient  number  of  equal  divisions  and  the  value  of  / 
determined  at  the  center  of  each  division.  The  reciprocal,  i, 
is  then  found.  It  is  convenient  to  use  the  same  number  of 
preliminary  divisions  as  is  desired  for  the  final  divisions.  The 
results  of  these  calculations  are  given  in  Table  E.  The  calcu- 


TABLE  E. 

DIVISIONS  OF  ARCH  RING. 


Properties  of  Preliminary  Equal  Divisions. 

Properties  of  Final  Divisions. 

No.  of 
Divi- 
sion. 

Depth, 
d 

Ic 

15/s 

/=/c  +  15/s 

i=f 

i 

ds 

/ 

d 

1 

2.55 

1.38 

.44 

1.82 

.549 

.560 

3.57 

1.78 

2.53 

2 

2.65 

1.55 

.48 

2.03 

.492 

.520 

3.85 

1.92 

2.59 

3 

2.75 

1.73 

.53 

2.26 

.442 

.480 

4.16 

2.08 

2.67 

4 

2.85 

1.93 

.57 

2.50 

.400 

.440 

4.53 

2.26 

2.75 

5 

2.95 

2.14 

.62 

2.76 

.362 

.405 

4  95 

2.47 

2.84 

6 

3.05 

2.36 

.68 

3.04 

.328 

.370 

5.42 

2.71 

2.93 

7 

3.15 

2.61 

.73 

3.34 

.300 

.333 

6.00 

3.00 

3.03 

8 

3.25 

2.86 

.79 

3.65 

.274 

.300 

6.67 

3.33 

3.15 

9 

3.35 

3.13 

.85 

3.98 

.251 

.265 

7.50 

3.75 

3.28 

10 

3.45 

3.42 

.91 

4.33 

.231 

.235 

8.51 

4.26 

3.42 

3.629 

] 

55  .  16 

3.63/10=  .363.     By  Eq.  (5)  ds  .  i= 


55.17X  .363 
10 


=2.00. 


lations  are  made  in  foot-units.  The  first  part  of  the  table 
relates  to  the  preliminary  ten  equal  divisions,  each  =  55. 17/10 
=  5.517  ft.  long.  The  resulting  values  of  i  are  plotted  in 
Fig.  103a.  The  line  AB  is  55.17  ft.  long  and  is  divided  into 
ten  equal  divisions,  1,  2,  3,  etc.  At  the  centers  of  the  several 
divisions  the  values  of  i  are  laid  off  as  ordinates,  ii,  i2,  i3,  etc., 


354 


ARCHES. 


CH.  VIII. 


§  185a.]  MAXIMUM    STRESSES.  355 

55  17 

and    the    curve    CD    drawn.     The    area    ABCD=  — ^-xSi 

=  55. 17  X  i'a.  This  area  is  now  to  be  divided  into  ten  equal 
parts,  each  equal  to  ds-i.  Each  of  these  parts  will  then  be 

equal  to  -~r^ — ?  =  2.00,  as  given  below  Table  E.     Beginning 

at  one  end  of  the  diagram,  Fig.  103a,  the  several  equal  areas 
are  then  laid  off,  the  values  of  i  being  scaled  from  the  diagram, 
and  ds  being  equal  to  2.00/1.  These  calculations  are  given 
in  the  latter  part  of  Table  E,  where  are  also  given  the  values 
of  7  and  d  for  the  center  points  of  the  final  subdivisions.  The 
diagram  of  Fig.  103a  is  not  really  necessary  but  is  here  intro- 
duced partly  to  make  clear  the  methods  employed  in  the 
calculations. 

Calculation  of  HQ,  F0,  and  MO  for  a  Load  Unity  at  Each  Load 
Point. — Fig.  1036  shows  the  arch  ring  with  the  subdivisions 
made  and  the  center  points  numbered  1,  2,  3,  etc.,  as  in  Fig. 
103.  The  load  points  considered  are  those  on  the  right,  A, 
B,  C,  D,  and  E.  These  are  the  center  points  of  the  supporting 
spandrel  walls.  [In  the  case  of  an  arch  with  continuous  load- 
ing, arbitrary  load  points  may  be  selected  sufficiently  close 
to  secure  the  desired  degree  of  accuracy  in  the  influence  lines. 
Four  of  five  such  points  will  generally  be  sufficient.]  The 
problem  now  is,  first,  to  calculate  H0,  F0,  and  M0,  for  a  load 
unity  placed  successively  at  these  several  load  points.  The 
computations  are  performed  in  the  same  manner  as  in  the 
preceding  examples.  Table  F  gives  all  the  calculations.  The 
quantities  m  are  equal  to  x  —  Xi,  where  x\  is  the  abscissa  of- 
the  load  point  in  question,  measured  from  the  crown.  For 
the  load  point  A  the  entire  load  is,  for  convenience,  assumed 
as  acting  just  to  the  right  of  the  crown  and  hence  as  belong- 
ing to  the  right  half.  The  shear  F0  is  evidently  equal  to  0.5 
in  this  case.  The  quantities  mL  do  not  appear.  For  load 
points  B,  C}  D,  and  E  the  calculations  become  progressively 
less  numerous. 


356 


ARCHES. 


[Cn.  VIII. 


TABLE  F. 
CALCULATIONS  FOR  HQ}  V0,  AND  M0. 


Load  at  A;   xi  =  0. 

Points. 

X 

V 

Z* 

1/2 

mR 

™Ry 

mRx 

1 

1.78 

0.02 

3.2 

.00 

1.78 

.0 

3.2 

2 

5.49 

0.29 

30.1 

.08 

5.49 

.8 

30.1 

3 

9.47 

0.62 

89.7 

.27 

9.47 

2.9 

89.7 

4 

13.76 

1.32 

189.3 

1.74 

13.76 

9.1 

189.3 

5 

18.38 

2.37 

337.7 

5.61 

18.38 

21.7 

337.7 

6 

23.34 

3.86 

544.7 

14.89 

23.34 

45.0 

544.7 

7 

28.67 

5.91 

821.8 

34.92 

28.67 

84.7 

821.8 

8 

34.37 

8.67 

1181.5 

75.10 

34.37 

148.9 

1181.5 

9 

40.44 

12.33 

1635.5 

151.93 

40.44 

249.3 

1635.5 

10 

46.82 

17.15 

2192.5 

294.08 

46.82 

401.5 

2192.5 

S 

52.54 

7026.0 

578.62 

-222.52 

-1927.9 

-7026.0 

TABLE  F — (Continued). 


Points. 

Load  at  B;  xi=10.0. 

Load  at  C;   zi  =  20.0. 

mR 

™RV 

mRx 

™R 

mRy 

mpx 

4 
5 
6 
7 
8 
9 
10 

3.76 
8.38 
13.34 
18.67 
24.37 
30.44 
36.82 

4.9 
19.8 
51.5 
110.3 
211.2 
375.2 
631.5 

51.7 
154.1 
311.2 
535.1 
837.6 
1231.3 
1724.3 

3.34 

8.67 
14.37 
20.44 
26.83 

12.9 
51.2 
124.6 
252.0 
460.0 

78.6 
248.4 
494.6 
826.6 
1255.2 

S 

-135.78 

-1404.4 

-4845.3 

-73.65 

-900.7 

-2903.4 

Points. 

Load  at  D;  x\  —  30.0. 

Load  at  E;  a*  =  40.0. 

mR 

mRv 

mRx 

mR 

MRV 

mRx 

8 
9 
10 

4.37 
10.44 
16.83 

37.9 

128.7 
288.5 

150.4 
422.5 
787.9 

0.44 
6.83 

5.4 
117.0 

17.8 
31.96 

V 

-31.64 

-455.1 

-1360.8 

-  7.27 

-122.4 

-337.4 

§  i85a.J  MAXIMUM    STRESSES.  357 

From  Table  F  we  then  have  for  the  denominators  of  eqs.  (1), 
(2),  and  (3)  of  Art.  174, 

2[(%)2-n22/2]  =  2[(52.54)2-10X578.62]= -6051.4, 
22x2  =  2x7026  =  14,052,       2n=20. 

The  values  of  H0,  F0,  and  MQ  then  result  as  follows: 
10  X  ( - 1927.9)  -  ( -  222.52)  X  52.54 


Load  at  A 


Load  at  B 


Load  at  C 


Load  at  D 


+1.254; 


-0.50; 


14,052 

-222.52  +  2X1.254X52.54 


+4.54. 


no— 


10  X  (  -  1404.4)  -  (-  135.78)  X  52.54 


-6051.4 


-4845.3 


14,052 


^=-0.345; 


-135.78  +  2X1.142X52.54 
Mo  =  -  -      oTT^ =  +  0.80. 


10X(-900.7)(-73.65)X52.54 


-6051.4 
-0.207; 


Load  at  E 


-2903.4 
14,052 
-  73.65  +  2  X  0.849  X  52.54 

20 

IPX  (-455.1)-  (-31.64)  X52.54 
-6051.4 

•3B?— •* 

-31.64+2X0.477X52.54 

20 

IPX  (- 122.4) -(- 7.27)  X  52.54 
-6051.4 

•T&4-0-024' 

-7.27  +  2X0.139X52.54 


358  ARCHES.  [Cn.  VIII. 

Calculation  of  Moments  and  Thrusts  at  any  Given  Section 
of  the  Arch  Due  to  Unit  Loads  at  the  Load  Points. — Consider,  for 
example,  a  load  unity  at  C,  Fig.  1036.  The  values  of  HQ,  V0t 
and  MQ  being  known,  the  moments,  shears,  and  thrusts  at 
any  given  section  can  now  be  found  either  graphically  or 
analytically,  as  already  explained.  The  graphical  method  is 
much  the  simpler.  For  this  purpose  draw  the  force  polygon, 
Fig.  103c,  the  load  AB  being  equal  to  unity,  and  lay  off  the 
shear  VQ=  —  .207  from  B}  upwards  to  C,  and  then  the  thrust 
#0  =  .849  to  the  right,  fixing  the  pole  0.  Then  from  the  center 
of  the  arch  at  the  crown  measure  down  a  distance  equal  to 

e=y^  =  -^— r  =.92    ft.,  fixing    one    point    on    the    equilibrium 

polygon.  Lines  KG  and  GL  drawn  parallel  to  OB  and  AO, 
intersecting  in  the  load  vertical  at  G,  complete  the  polygon. 
The  other  polygons  are  drawn  in  a  like  manner. 

Having  these  polygons  drawn,  the  bending  moment  at  any 
section,  due  to  any  one  of  the  unit  loads,  is  equal  to  the  vertical 
ordinate  measured  from  the  proper  equilibrium  polygon  to 
the  gravity  axis  of  the  arch,  multiplied  by  the  corresponding 
pole  distance.  From  these  several  polygons  the  moments  can 
therefore  be  found  at  any  section  for  a  unit  load  at  any  load 
point  on  either  half  of  the  arch,  and  the  influence  line  drawn 
for  such  moment.  The  tangential  thrusts  can  likewise  be 
determined. 

Influence  Lines  for  Fiber  Stresses  at  any  Section. — Instead  of 
constructing  influence  lines  for  moment  and  thrust  it  will  be 
more  direct  to  construct  them  at  once  for  stress  on  extreme 
fiber. 

From  Art.  82  the  stress  on  extreme  fiber  is 

Mu    N 
T+A> 

in  which  u  is  the  distance  from  the  neutral  axis  to  the  fiber 
in  question  and  N  is  the  thrust  normal  to  the  section.  The 


§  I85a.]  MAXIMUM    STRESSES.  359 

bending  moment  is  equal  to  the  thrust  times  its  eccentricity, 
or  M  =  Ne. 

If  r  =  radius  of  gyration  of  the  section,  we  have  Ar2  =  L 
The  above  expression  for  fc  may  then  be  written  in  the  form, 

N--U 
Neu        u 

~T    ~T~ 
ti(e+- 


j         (16) 

r2  I      r2\ 

The  quantity  —  is  a  length,  and  N(e-\ — )    is    a    moment 

which  may  be  written  M' '.     Then  we  have 

/=^ (17) 

This  new  moment,  M',  is  equal  to  the  thrust  N,  multiplied 

r2 

by  the   eccentricity  e  plus  the   additional  distance  — .    We 

r2 

may  then  compute  the  value  of  --  for  the  section  considered 

and  take  the  center  of  moments  at  this  distance  above  or 
below  the  neutral  axis,  according  as  the  lower  or  upper  fiber 
stress  is  desired.  By  so  selecting  the  center  of  moments  the 
fibre  stress  becomes  equal  to  the  moment  multiplied  by  the 

usual  factor  j,  and  therefore'  varies  with  the  moment.    An 

influence  line  drawn  for  such  moment  will  therefore  serve  as 
an  influence  line  for  fiber  stress. 

In  this  problem  influence  lines  have  thus  been  drawn  for 
upper  and  lower  fiber  stress  at  sections  taken  at  the  various 
load  points  and  at  the  springing  line  F.  They  are  given  in 
Fig.  103d.  To  explain  further  their  construction  consider 
the  section  at  D,  and  suppose  the  arch  is  loaded  with  a  unit 
load  at  C.  The  arch  near  D  is  shown  to  a  larger  scale  in 
Fig.  103e.  The  force  R  is  the  resultant  of  the  forces  on  the 


360 


ARCHES. 


[Cn.  VIII. 


SECTION  A  / 


SECTION  B 


SECTION  C  / 


SECTION  D  f 


SECTION  E 


SECTION  F 


FIG.  103d. 


§  185o.]  MAXIMUM    STRESSES.  361 

left  and  acts  on  the  line  GL,  Fig.  1036.     In  amount  it  is  equal 
to  AO  in  the  force  polygon.     The  neutral  axis  is    at  a,  Fig. 
103e,  and  the  distances  abi  and  ab^ 
are  the  values  of  r2/u  calculated 
for  this  section.     The  point  bi  is 
therefore    the  center  of   moments 
for  Mf  for  upper  fiber  stress  and 
b2   is   the   center  of   moment   for 
M'   for     the     lower   fiber    stress. 
(The  points  61   and  b2  are  at  the 
edges  of  the  "  kern  "  of  the  section. 
In    a   simple    rectangular   section 
they  are  at  the  edges  of  the  mid- 
dle third  and  in  a  truss  or  plate  girder,  where  the  flange  takes 
all  the  moment,  they  are  at  the  flange  centers.) 

These  values  of  M'  are  given  graphically  by  RxbiCi  and 
RXb2c2,  or  by  the  more  convenient  products  #xMi  and 
Hxb2d2.  These  moments  are  calculated  and  plotted  in  Fig. 
103d  at  point  C,  under  the  load  unity.  They  are  the  ordi- 
nates cc\  and  cc2,  and  are  plotted  as  positive  ordinates  since 
they  represent  positive  bending  moments.  In  the  same  manner 
values  of  the  moments  are  determined  at  section  D  for  unit 
loads  at  A,  B,  D,  and  E,  and  plotted  at  corresponding  points 
in  the  diagram.  For  loads  on  the  left  half  of  the  structure 
the  values  are  found  by  considering  the  section  D  to  be  on 
the  left  half  of  the  arch  at  D',  and  measuring  the  several 
ordinates  to  the  reaction  lines  as  drawn  for  loads  on  the  right. 
The  complete  influence  lines  are  given  in  the  diagram.  The 
upper  line  f'aid2f  is  for  upper  fiber  stress  and  the  lower  line 
f'a2d2f  is  for  lower  fiber  stress.  The  dotted  line  (midway 
between,  for  symmetrical  sections)  is  the  influence  line  for  the 
usual  bending  moment,  center  of  moments  at  the  neutral 
axis.  Since  positive  ordinates  indicate  positive  moments  it 
follows  that  for  the  upper  fibers  positive  ordinates  indicate 
compression  and  for  the  lower  fibers  positive  ordinates  indi- 
cate tension.  Negative  ordinates  indicate  respectively  tension 


362 


ARCHES. 


[Cn.  VIII. 


on  the  upper  fibers  and  compression  on  the  lower.  The  actual 
fiber  stress  at  section  D  due  to  a  unit  load  at  any  point  is  now 
given  by  the  ordinate  to  the  influence  line  at  the  point,  mul- 
tiplied by  y  for  this  section. 

In  Table  G  are  given  values  of  u,  7,  r2,  and  --  for  each 

u 

of  the  sections  A  .  .  .  F.  The  distances  r2/u  are  shown  by  the 
dotted  lines  in  the  arch  ring,  Fig.  103&. 

TABLE  G. 
PROPERTIES  OF  SECTIONS  A  .    .  F, 


Sec- 
tion. 

Depth,  d. 

w  =  id. 

/ 

A 

(Ae+15Aa) 

r2 

r2 
u 

A 

2.50 

1.25 

1.70 

2.92 

.58 

.46 

B 

2.78 

1.39 

2.10 

3.20 

.65 

.47 

C 

3.00 

1.50 

2.54 

3.42 

.74 

.49 

D 

3.18 

1.59 

3.06 

3.60 

.85 

.53 

E 

3.35 

1.67 

3.70 

3.77 

.98 

.59 

F 

3.50 

1.75 

4.46 

3.92 

1.14 

.65 

Where  the  arch  is  continuously  loaded  the  influence  lines 
should  be  drawn  as  smooth  curves  through  the  points  deter- 
mined by  the  ordinates  at  the  load-points  selected. 

Maximum  Fiber  Stress  at  Any  Section. — Having  the  influence 
lines  drawn  for  several  sections,  as  in  Fig.  103d,  the  maximum 
fiber  stress  at  the  various  sections  can  readily  be  determined 
for  any  given  loading,  either  uniform  or  concentrated.  In 
the  figure  the  various  positive  and  negative  areas  above  and 
below  the  axis  have  been  measured  and  are  written  within 
the  respective  areas.  The  upper  figure  refers  in  all  cases  to 
the  area  between  the  axis  and  the  upper  curve.  For  uniform 
loads  these  areas,  multiplied  by  the  load  per  foot,  give  the 
respective  moments.  For  concentrated  loads  the  moments  are 
found  by  summing  the  products  of  the  several  loads  times 
the  corresponding  ordinates.  The  maximum  values  can 
readily  be  determined  by  trial.  The  influence  lines  show  very 
clearly  the  extent  and  general  position  -of  loads  for  maximum 
stresses. 

For  example,  at  the  crown,  section  A,  the  moment  for  upper 


§186.]  MAXIMUM    STRESSES.  363 

fiber  stress,  due  to  a  uniform  dead  load  of,  say,  800  pounds 
per  foot,  =  [76.4  -  (2  X  12.2)]  X  800  =  41,600  ft.-lbs.     The  stress 

lbs/ft2,  =330  lbs/in2.     For  a 


uniform  live  load  of  500    Ibs.  per  foot,  the  maximum  stress  is 
caused  when  the  load  extends  from  K  to  L,  and  the  stress  for 

such  load  =  76.4x500X^-144  =  195  lbs/in2. 

Again,  at  section  (7,  the  dead  load  upper  fiber  stress  = 

1  50 
(111.7-61.  1)X800X^H-144  =  166  lbs/in2;     and    the   maxi- 


1  50 
mum  live  load  stress  =  111.  7  X  500  X  ^7  -144  =  230  lbs/in2,  the 

Z.o4 

load  extending  from  K  to  /.  For  the  maximum  lower  fiber 
stress  the  load  extends  from/'  to  L  and  the  stress  =  237  lbs/in2; 
the  dead  load  stress  =  120  lbs/in2.  Othe  rstresses  are  found 
in  a  similar  manner. 

The  influence  lines  show  that  in  arches  of  proportions  such 
as  here  considered,  the  loading  for  maximum  stresses  is  about 
as  noted  in  Art.  185.  For  the  crown,  the  center  third  should 
be  loaded;  for  the  haunch  (sections  C  and  D)  the  load  should 
extend  over  about  two-thirds  or  one-third  the  span  length, 
and  about  the  same  for  the  section  at  the  springing  line. 

Case  in  which  the  Resultant  Stress  is  Tensile.  —  If  the  resultant 
stress  is  tensile  at  any  section  and  the  tension  in  the  concrete 
is  to  be  neglected  then  the  formulas  for  fiber  stress  of  Art.  83 
must  be  used,  which  makes  it  necessary  to  plot  the  influence 
lines  for  the  true  bending  moment  and  to  determine  the  thrust 
as  well  as  the  moment.  The  diagram,  Plate  XIV,  p.  288, 
can  then  be  used. 

1  86.  Illustrative  Examples  of  Arch  Design.  —  Fig.  104 
shows  a  longitudinal  section  and  part  plan  of  a  bridge  at  Grand 
Rapids,  Mich.*  It  consists  of  five  spans  of  lengths  from  79  to 
87  feet.  The  reinforcement  is  composed  of  IJ-in.  Thacher  bars 
spaced  14  in.  apart  near  both  the  extrados  and  intrados.  Each 
pair  is  connected  by  f-in.  connecting-rods  spaced  4  ft.  apart. 

*  Eng.  News,  Vol.  LII,  1904,  p.  490. 


364 


ARCHES. 


[Cn.  VIII. 


PART  PLAN  5; 

Top  of  Earth  Fill      *y    x  Curb  Grade  Line 


LONGITUDINAL  SECTION 

Fie.  104,— Arch  Bridge  at  Grand  Rapids,  Mich. 


FIG.  105. — ^Arch  Bridge  on  the  Chicago  &  Eastern  Illinois  R.R. 


186.] 


EXAMPLES    OF   ARCH   DESIGN. 


365 


Fig.   105  illustrates  a  38-ft.  span  reinforced  arch  on  the 
C.  &  E.  I.  R.R.*    Johnson  corrugated  bars  were  used. 


Figs.  106  and  107  show  the  details  of  a  design  for  a    concrete 
viaduct  at  Milwaukee,  Wis.f    The  design  was  made  by  the 

*  R.R.  Gaz.,  April  13,  1906,  p.  390. 
t  Eng.  News,  Vol.  LVII,  1907,  p.  178. 


366 


ARCHES. 


[Cfl.  VIII. 


Concrete-Steel  Engineering  Co.,  of  New  York  City,  and  is  a 
typical  Melan  arch.  The  reinforcement  consists  of  built-up 
ribs  of  3"X3"Xi"  angles  connected  by  lattice  bars  2"Xj". 


l-si^., 

f   i  TO'O"    -  e 

4  "Wood  Blocks 
/  J$'  Cushion 

hs/6Y5/61    ••«—  * 

Slope  M=12     \ 

20  Spacer  at  3'0=60/0l- 


HALF  VERTICAL  SECTION 

,AT  CENTER  LINE 

OF  PIER 


HALF  VERTICAL  SECTION  A-B 


FIG.  107. 


The  ribs  are  spaced  3  ft.  apart.  These  ribs  are  designed  to 
carry  the  entire  bending  moment  at  a  stress  of  18,000  lbs/in2. 
The  stress  in  the  concrete  was  limited  to  500  lbs/in2  in  com- 


§  186.] 


EXAMPLES    OF    ARC  i    DESIGN. 


367 


ARCHES 


[CH.  VIII. 


pression,  or  600  lbs/in2  including  temperature  stresses.  The 
roadway  is  supported  over  a  considerable  portion  of  the  span 
length  by  means  of  a  reinforced  floor  carried  on  vertical  walls. 


Hoops 
HALF  SECTION  OF  CROWN 


HALF  SECTION  AT  PIER 
FIG.  109. 

The  viaduct  contains   eight  spans  of  the  dimensions  shown  in 
the  illustration. 

Figs.  108  and  109  illustrate  another  design  for  the  same  via- 
duct mentioned  in  the  preceding  paragraph.*    This  design  was 

*  Eng.  News,  Vol.  LVII,  1907,  p.  178. 


186.] 


EXAMPLES    OF    ARCH    DESIGN. 


369 


submitted  by  Mr.  C.  A.  P.  Turner,  of  Minneapolis,  Minn.  In 
this  case  the  arch  is  composed  of  three  ribs  4  ft.  9  in.  square  at 
the  crown  and  7  ft.  3  in.  square  at  the  pier.  The  rib  reinforce- 
ment is  composed  of  longitudinal  rods  arranged  in  a  circle  and 
connected  at  frequent  intervals  by  bands  of  2$"Xf"  metal. 
The  stirrups  and  bent  bars  in  the  floor-beams  and  slabs  give  a 
very  effective  reinforcement. 


CHAPTER  IX. 

RETAINING-WALLS  AND  DAMS. 

187.  Advantages    of   Reinforced    Concrete. — Retaining- 
walls,  dams,  bridge  abutments,  and  the  like  constitute  a  class  of 
structures  in  which  the  outside  forces  acting  are  mainly  hori- 
zontal, and  in  which,  therefore,   the  question  of  stability  is 
largely  a  question  of  safety  against  overturning.    Where  ordi- 
nary masonry  is  used  in  these  structures  the  weight  of  the 
material  must  be  depended  upon  to  balance  the  overturning 
forces;  for  though  the  structure  be  anchored  to  the  foundation 
no  tensile  stresses  can  be  allowed  in  the  masonry.    As  a  con- 
sequence of  these  limitations  the  maximum  compressive  stresses 
in  such  structures  are  not  high,  except  in  extreme  cases,  so  that 
generally  the  dimensions  are  determined  by  the  weight  of  the 
material.     The  application  of  reinforced  concrete  in  such  cases 
enables  the  design  to  be  so  modified  as  to  utilize  the  weight  of 
the  material  to  be  retained  as  part  of  the  resisting  weight  and 
to  calculate  the  sections  to  develop  more  nearly  the  full  strength 
of  the  concrete.     A  very  considerable  gain  in  economy  therefore 
results. 

RETAINING-WALLS. 

188.  Method  of  Determining  Stability.— No  attempt  will 
be  made  here  to  present  the  various  mathematical  theories  of 
earth  pressure.    Unless  the  results  obtained  from  such  theories 
are  carefully  controlled  by  the   results  of   experience  they  are 
apt  to  be  very  misleading.     Probably  the  most  satisfactory  way 
to  design  a  reinforced  concrete  retainirig-wall,  as  regards  sta- 
bility against  overturning,  is  to  proportion  it  so  that  it  will  be, 

370 


§  189.]          FLUID    PRESSURE   FOR    MASONRY    WALLS.  371 

as  nearly  as  possible,  equivalent  to  a  solid  masonry  wall  of  such 
a  section  as  is  known  to  have  given  satisfactory  results  under 
the  given  conditions.  Rules  of  practice  as  to  solid  masonry 
walls  have  long  been  established.  They  represent  the  accu- 
mulated experience  of  many  engineers  and  are  based  upon  data 
obtained  from  many  failures  as  well  as  from  successful  designs. 
Until  experience  is  had  directly  with  the  reinforced  type  of  wall 
its  stability  may,  therefore,  well  be  determined  by  comparison 
with  the  older  form  of  construction.  The  analysis  given  here 
will  consequently  be  limited  to  a  convenient  method  of  com- 
parison of  the  two  types.  It  may  be  said  in  passing  that  good 
construction  requires  quite  as  much  attention  to  the  earth 
filling  itself  and  to  its  drainage  as  to  the  design  and  construc- 
tion of  the  wall. 

In  dimensioning  a  reinforced  concrete  wall  which  will  possess 
stability  equal  to  that  of  a  given  solid  wall,  it  will  be  convenient 
to  determine  the  equivalent  fluid  pressure  under  which  the 
solid  wall  will  be  stable  and  then  apply  this  pressure  to  the 
reinforced  type  of  wall.  The  basis  of  the  calculation  of  this 
fluid  pressure  will  be  to  determine  the  weight  per  cubic  foot  of  a 
fluid  which  will  exert  such  a  pressure  against  the  solid  wall  as 
to  cause  the  resultant  of  all  forces  above  the  base  to  intersect 
the  base  at  the  edge  of  the  middle  third.  If,  then,  the  reinforced 
wall  be  designed  so  that  it  will  be  equally  stable  against  this 
pressure,  it  will  be  practically  equivalent  to  the  solid  wall. 

It  will  be  seen  that  this  method  is  very  simple  and  adapts 
itself  readily  to  the  utilization  of  present  rules  of  practice.  If 
desired,  the  theory  of  earth  pressure  may  of  course  be  directly 
applied  to  the  problem. 

189.  Equivalent  Fluid  Pressure  for  Ordinary  Masonry 
Walls. — Two  forms  of  wall  will  be  considered  (Fig.  110).  Form 
(a)  is  the  more  common  form  of  wall.  A  small  batter  is  usu- 
ally given  to  the  front  face,  and  the  back  face  is  sloped  in  an 
irregular  line,  the  width  of  the  top  being  as  narrow  as  circum- 
stances may  warrant.  Such  a  wall  will  be  stable  when  the 
width  of  the  base  is  made  from  one-third  to  one-half  the  height, 


372 


RETAINING-WALLS   AND    DAMS. 


[Cn.  IX. 


four- tenths  being  a  common  rule  of  practice.  Form  (6)  is  used 
for  relatively  low  walls.  Its  width  may  be  a  little  less  than  that 
of  form  (a)  for  equal  stability.  While  the  calculations  here 
given  apply  only  to  the  two  forms  as  represented  in  Fig.  110, 
the  results  will  be  but  little  different  for  walls  similar  in  form 
but  which  vary  considerably  therefrom. 

Form  (a). — The  height  is  h  and  the  bottom  width  I.    The 
batter  of  the  front  face  will  be 
taken  at  1:12,    and    the    top  ~* 
width  at   1/6   of   the  bottom 
width.      The    weight    of    the 
masonry  will   be  assumed   at  ;' 
150,    and    that   of   the   earth 
filling  at  100  lbs/ft:{.    It  will 
be  assumed  that  the  fluid  pres- 
sure   acts    against   a  vertical          .     ,  > 
plane  FC]  the  stability  of  the  FjG<  HQ 

entire  volume  to  the  left  of  this 

plane,  including  the  weight  of  the  earth,  will  be  determined. 
Let  Wi  denote  the  weight  of  masonry  per  lineal  foot,  and  W2 
the  weight  of  the  earth  filling  to  the  left  of  FC.  Let  P  denote 
the  resultant  fluid  pressure  acting  at  a  distance  J/i  above  the 
base.  Let  p  denote  the  weight  per  cubic  foot  of  such  fluid. 


w, 


Assume  that  the  resultant  pressure  due  to  the  weight  of  the 
wall  TFi,  the  weight  of  the  earth  W?,  and  the  pressure  P,  inter- 
sects the  base  at  the  edge  of  the  middle  third.  Equating  mo- 
ments about  this  point  we  derive  the  relation 


(1) 


Form  (ft). — In  this  form  the  only  forces  to  be  considered  are 
the  weight  Wl  -and  the  pressure  P.    Equating  these  as  before 

there  results 

72 

...     (2) 


§  190.] 


REINFORCED-CONCRETE  WALLS. 


373 


If  the  top  width  in  form  (a)  be  made  zero  the  effect  on  the 
result  would  be  to  change  the  coefficient  in  Eq.  (1)  from  132  to 
127,  thus  showing  that  a  considerable  variation  in  top  width 
has  little  effect  on  the  result. 

Substituting  various  values  of  l/h  in  Eqs.  (1)  and  (2)  we  have 
the  following  values  of  p,  or  equivalent  fluid  weight,  under  wThich 
the  wall  is  stable  as  above  assumed. 

l/h  P 

1/3  14.71bs/ft8 

Form  (a)    1     4/10  21.1      " 

[     1/2  33.0      " 


Form  (6) 


1/4 
1/3 
4/10 


9.4 
16.7 
24.0 


According  to  these  calculations  a  fluid  weight  of  20  to 
25  lbs/ft3  may  be  taken  as  a  basis  of  design  to  secure  sta- 
bility equivalent  to  the  ordinary  wall,  assuming  the  resultant 
pressures  to  cut  the  edge  of  the  middle  third  and  counting  the 
weight  of  earth  vertically  above  the  back  slope  as  part  of  the 
resisting  load.  It  is  to  be  noted  that  the  pressures  herein 
determined  are  not  necessarily  the  actual  earth  pressures;  the 
results  are  to  be  used  only  as  a  means  of  securing  stability  of 
reinforced  walls  approximately  equal  to  that 
of  solid  walls  of  known  proportions. 

190.  Stability  of  Reinforced  Concrete 
Walls. — Fig.  Ill  represents  in  outline  the 
usual  type  of  reinforced  wall.  It  consists  of 
a  vertical  wall  AE  attached  to  a  floor  DC. 
For  low  walls  the  upright  part  AE  may  act 
simply  as  a  cantilever;  and  likewise  the  parts 
EC  and  ED.  For  larger  walls  the  part  AE  is 
tied  to  EC  at  intervals  by  back  walls  ACE  in 
the  form  of  narrow  transverse  walls  with  ten- 
sion reinforcement.  The  projecting  portion  ED  may  still  act 
as  a  cantilever,  or  it,  also,  may  be  connected  to  the  vertical 


FIG.  111. 


374  RETAINING-WALLS    AND    DAMS.  [Cn.  IX. 

wall  AE  by  means  of  buttresses.  In  either  case  the  earth 
pressures  act  in  essentially  the  same  manner  and  the  necessary 
width  of  base  is  found  in  the  same  way. 

Let  Z=  width  of  base; 

x  =  distance  from  toe  to  back  of  wall  AE] 
/*=  height; 

p=equivalent  fluid  weight  as  determined  in  Art.  189; 
w2=  weight  of  earth  filling  per  cubic  foot; 
Wi  =  weight  of  masonry  per  lineal  foot; 
W2  =  weight  per  lineal  foot  of  earth  above  the  floor  EC; 
a  Clever-arm  of  Wi  about  point  F,  the  edge   of   the 

middle  third  ; 
P=  total  fluid  pressure  = 


Then  equating  moments  about  the  point  F  we  have 

,    .....    (3) 


or 

Wia+wM-x)(v-l-$=^  .....    (4) 

If  the  wall  AE  is  placed  well  towards  the  front  the  moment 
of  the  masonry  will  be  small.  Neglecting  this  term  and  putting 
x=kl  we  may  solve  for  I,  getting 


This  is  a  minimum  for  k  =  £,  that  is,  for  x  =  JZ.    With  this 
value  of  k  we  have 


/-.  87-.  fc  ........     (6) 

For  w  =  100 

l=.087Vp.h  .......    (7) 

If,  for  example,  the  value  of  p  be  taken  at  21.1,  correspond- 
ing to  a  value  of  l/h  =  4/W  for  a  solid  wall,  the  value  of  /  is 


§  190.] 


REINFORCED-CONCRETE    WALLS. 


375 


equal  to  .087XV21.1X/&  =  .4/&,  or  the  same  as  the  width  of  the 
solid  wall. 

As  it  may  be  desirable  to  use  a  smaller  or  larger  value  of  x 
than  J/,  Table  No.  22  has  been  prepared  giving  the  values  of 
l/li  for  various  values  of  x/l  and  various  values  of  p.  An  ex- 
amination of  the  table  shows  plainly  that  the  length  of  the  pro- 
jection x  makes  very  little  difference  in  the  required  total 
length  of  base.  However,  with  x  made  very  small  or  very 
large  the  weight  of  the  wall  should  be  taken  into  account.  A 
further  fact  brought  out  by  the  table  and  by  the  table  of  Art. 
189  is  that  the  stability  of  the  reinforced  wall  is  about  the  same 
as  a  solid  wall  of  form  (a)  shown  in  Fig.  110  and  having  the 
same  base  length. 

TABLE  No.  22. 

PROPORTIONS  OF  REINFORCED-CONCRETE  RETAINING-WALLS. 

(See  Fig.  111.) 

VALUES  OF  l/h  FOR  DIFFERENT  VALUES  OF  p  AND  FOR  w2=100  (Eg.  (5)). 


Values  of 

L.          ~/1 

Values  of  Equivalent  Fluid  Weight  p.     Pounds  per  Cubic  Foot. 

15 

20 

25 

33 

.5 

.35 

.40 

.45 

.51 

.33 

.34 

.39 

.43 

.50 

.25 

.34 

.39 

.44 

.50 

.20 

.34 

.40 

.44 

.51 

.15 

.35 

.40 

.45 

.52 

JO 

.36 

.41 

.46 

.53 

0 

.39 

.45 

.50 

.57 

The  resultant  forces  acting  upon  the  three  parts  of  the  wall 
AE,  DE,  and  EC  must  be  determined.  On  the  wall  AE  the 
force  may  be  taken  as  a  horizontal  force  equal  to  P,  =  %ph2,  and 
applied  a  distance  %h  above  the  base.  The  resultant  force 
acting  on  any  length  h'  from  the  top  is  likewise  %ph'2  and  ap- 
plied a  distance  f h'  below  the  top.  The  pressure  on  the  founda- 
tion will  equal  the  total  weight  TFi  +  W2  and  will  be  appliel  a 
distance  JZ  from  point  D.  The  average  unit  pressure  will  be 


376  RETAINING-WALLS    AND    DAMS.  [Cn.  IX. 

(Wi  +  W2)/l,  and  the  maximum  pressure  at  D  will  be  twice 
this  value. 

The  upward  pressure  under  the  cantilever  DE  will  vary  from 

a  maximum  of  2  —  ^—.  —  *  at  D  to  a  value  under  the  point  E  of 

2  —  —j  —  "*~~7~"     ^is   k   a   "  trapezoid"    of    pressure,    and 

where  x  is  large  the  centre  of  gravity  of  the  trapezoid  may  be 
found  and  the  resultant  applied  at  this  point.  Usually  it  will 
be  accurate  enough  to  assume  the  pressure  on  DE  as  uniformly 
distributed  at  an  average  value  and  applied  at  the  centre  of 
the  projection  outside  of  the  vertical  wall. 

The  upward  pressure  on  the  floor  EC  varies  from  the  value 
above  given  at  E,  to  zero  at  C.  It  varies  uni- 
formly between  these  points.  The  downward 
pressure  is  the  weight  of  the  earth  above  the 
floor,  =TF2.  This  may  be  assumed  as  uniformly 
distributed  and  equal  to  v^h  per  unit  area  at  all 
points.  The  total  downward  pressure  on  EC  will 
be  greater  than  the  upward  pressure  unless  x  is 
very  small. 

191.  Design  of  Wall.—  In  discussing  the  design  it  will  be 
necessary  to  consider  two  forms:  (1)  the  cantilever  wall  without 
back  tie-walls  as  in  Fig.  112,  and  the  wall  provided  with  such 
back  w^alls  as  in  Fig.  113. 

The  form  of  Fig.  112  is  adapted  to  heights  of  about  12  to 
18  feet.  For  high  walls  the  form  of  Fig.  113  will  be  more 
economical. 

Form    (a).  (Fig.    112.)—  The    maximum    moment    in    the 

h    ph? 
upright  portion  AE  is  P7>=~a~-    At  any  distance  hf  below 


the  top  the  moment  is  ^7—.    Only  a  portion  of  the  reinforcing- 

rods  need  be  carried  up  the  full  height.    The  shear  at  the  bot- 

vh2 
torn  is  P—*|r.    This  will  be  very  small  and  will  require  no 


§  191.]  DESIGN   OF   WALL.  377 

special  attention.  The  reinforcing-rods  of  a  cantilever  beam 
have  their  maximum  stress  at  the  end  of  the  beam,  hence  special 
care  must  be  given  to  secure  an  effective  bond  or  anchorage. 
In  the  figure  the  vertical  rods  have  an  insufficient  length  below 
the  point  of  maximum  moment  to  develop  their  full  strength, 
and  therefore  they  should  be  anchored  in  a  substantial  man- 
ner. This  may  be  done  by  screw-ends  and  nuts,  or  by  loop- 
ing the  rods  around  anchor-bars  near  the  bottom  of  the  floor 
DC. 

The  cantilever  DE  must  be  treated  in  the  same  manner  as 
the  upright  cantilever.     The  pressures  will  be  much  heavier  • 
and  the  shear  and  bond  stress  may  need  attention.     The  rein- 
forcement should  extend  far  enough  beyond  E  for  bond  strength. 

The  cantilever  EC  is  acted  upon  by  an  upward  and  a  down- 
ward force  as  shown  in  the  figure.  The  maximum  moment  will 
be  at  E  and  will  be  negative.  It  is  provided  for  by  reinforce- 
ment as  shown. 

To  secure  maximum  economy  each  one  of  the  cantilevers 
may  be  tapered  towards  the  end  to  a  minimum  practicable 
thickness.  The  bending  moments  at  various  sections  in  a 
cantilever  beam  uniformly  loaded  vary  as  the  squares  of  the 
distances  from  the  free  end.  The  resisting  moments  vary  ap- 
proximately as  the  squares  of  the  depths  of  the  beam.  Hence  a 
beam  tapering  uniformly  to  zero  depth  at  the  end  would  be  of 
the  necessary  depth  at  all  points.  The  moments  in  the  vertical 
beam  AE  vary  as  the  cubes  of  the  distances  below  the  top,  so 
that  a  straight  taper  will  in  this  case  give  a  beam  whose  weakest 
point  will  be  at  the  bottom.  At  the  top  point  A  some  form  of 
coping  is  usually  added,  of  a  width  according  to  the  require- 
ments of  the  case. 

To  prevent  unsightly  cracks  a  certain  amount  of  longitudinal 
reinforcement  is  necessary.  The  amount  required  per  square 
foot  of  cross-section  will  be  less  the  heavier  the  wall,  as  tem- 
perature changes  will  be  less  m  such  a  wall.  On  the  basis  of 
the  discussion  in  Chap.  V,  Art.  142,  the  percentage  required 
may  be  placed  at  about  0.4%  as  a  maximum  for  thin  walls,  to 


378 


RETAINING-WALLS    AND    DAMS 


[Cn.  IX. 


perhaps  one-half    of  this  for  heavy  walls.     High  elastic-limit 
material  is  advantageous  for  this  purpose. 

Form  (b).  (Fig.  113.) — So  far  as  the  external  pressures  are 
concerned  they  have  been  explained  in  Art.  189,  and  are  prac- 
tically the  same  as  in  the  previous  case  considered.  The  loads 
or  pressures  on  the  concrete  are,  however,  carried  quite  differ- 
ently. The  toe  DE  is  the  same  as  in  form  (a)  and  reinforced 
in  the  same  wTay.  The  pressure  against  the  longitudinal  wall  AE  is 
carried  laterally  for  the  most  part  and  given  over  to  the  inclined 


j 


(a) 

CROSS-SECTION 


(6) 

ELEVATION 

FIG.  113. 


A 

II 

A 

1  | 
\.4- 

i    ! 

I 
1 

i 

If 
i 

C 

C' 

PLAN 


back  walls.  The  wall  AE  must  therefore  be  designed  as  a  slab 
supported  along  the  lines  AE  and  A'E'  (Fig.  113  (6)),  and  sub- 
jected to  a  pressure  per  square  foot  at  any  point  a  distance  h' 
belowr  the  top  equal  to  phr.  Near  the  bottom,  the  load  on  AE 
is  transmitted  more  or  less  to  the  floor  EC.  The  wall  should 
therefore  be  bonded  to  the  floor  with  a  small  amount  of  vertical 
reinforcement,  which  may  well  extend  to  the  top  to  prevent 
cracks,  although  under  ordinary  conditions  the  wall  AE  is  under 
some  vertical  pressure. 

The  floor  EC  is  subjected  to  both  upward  and  downward 
pressures,  the  latter  exceeding  the  former  towards  the  end  C, 


§191] 


DESIGN    OF   WALL. 


379 


and  possibly  throughout,  as  previously  explained.  This  floor 
is  supported  by  the  back  wall  A  EC  and  is  therefore  reinforced 
longitudinally  as  a  floor-slab  in  accordance  with  the  resultant 
pressure  at  any  point.  Here,  again,  it  is  well  to  bond  the  floor 
to  the  wall  AE  by  extending  the  transverse  reinforcement  of 
the  toe  DE  into  the  portion  EC. 

The  back  wrall  ACE  acts  as  a  cantilever  beam  anchored  to 
the  floor.  It  is  also  a  T-beam,  the  flange  being  the  longi- 
tudinal wall  AE.  The  tension  along  the  edge  AC  is  carried  by 
rods  near  this  edge,  wrhose  stress  at  any  point  is  found  with 
sufficient  accuracy  by  an  equation  of  moments  taken  about  the 
center  of  the  front  wall.  The  maximum  stress  will  be  at  the 
bottom,  if  the  wall  is  made  with  a  straight  profile.  At  the 
connection  of  the  wall  AEC  to  the  floor,  it  is  to  be  noted  that 
the  floor  load  is  transferred  to  the  wall  along  the  line  EC, 
but  mainly  near  the  end  C.  The  main  tension-rods  in  AC 
should  therefore  be  distributed  somewhat  at  their  lower  ends 
and  well  anchored  to  the  reinforcing-rods  of  the  floor  EC.  A 
few  additional  vertical  rods  should  also  be  put  in  to  insure 
thorough  bonding  of  floor  to  wall.  These  will  also  carry  a  part 
of  the  tension  in  the  back  wall,  but  will  not  be  as  efficient  as 
A^  the  rods  nearer  the  outside  edge.  It  is  desira- 

ble, likewise,  to  bond  the  vertical  wall  AE  to  the 
back  wall  with  short  horizontal  rods  as  show^n. 
The  slabs  formed  by  the  walls  AE  and  the 
floor  EC  are  continuous  over  supports,  and  if 
the  span  is  long  should  be  provided  with  some 
reinforcement  for  negative  moments  at  these 
supports. 

Fig.  114  shows  some  additional  features  of 
design  which  have  been  used.    A  longitudinal 
beam  is  built  at  C  and  the  floor  is  thus  supported 
on  all  four  edges.     The   main  rods  along  AC 
are  then  anchored  into  the  beam. 

A  horizontal  beam  may  also  be  made  of  the  coping  at  A, 
thus  giving  some  support  to  the  wall  A B  along  its  upper  edge. 


u     "   "'    i 


FIG.  114. 


380 


RETAINING-WALLS    AND    DAMS. 


[Cn.  IX. 


A  projection  may  be  necessary  at  the  toe  7),  or  elsewhere,  in 
order  to  increase  the  resistance  against  forward  sliding.  The 
beam  C  aids  in  this  respect. 

192.  Illustrative  Examples.— Fig.  115  shows  the  form  of 
retaining-wall  used  on  the  Great   Northern  R.R.  at  Seattle, 


SE-CTLQN 


FLAN 


FIG.  115. — Retaining-wall,  Great  Northern  Railway. 

Wash.*  This  is  a  good  illustration  of  the  second  type  above 
discussed.  An  estimate  by  Mr.  C.  F.  Graff  of  the  amounts  of 
material  per  lineal  foot  required  in  reinforced  and  plain  con- 


*  Eng.  News,  Vol.  LIII,  1905,  p.  262. 


ILLUSTRATIVE    EXAMPLES. 


381 


I 
1 
I 


I 


382 


RETAINING-WALLS   AND  DAMS. 


CH.  IX. 


crete  walls,  made  in  connection  with  the  design  of  Fig.  115,  gave 
the  following  results: 


Amount  of  Concrete  per  Lineal  Foot. 

Height  of  Wall,  Feet, 

Saving:    Per  Cent,  of 
Reinforced  Wall. 

Plain  Wall, 

Reinforced  Wall, 

Cubic  Feet. 

Cubic  Feet. 

40 

396.4 

218 

45 

30 

226 

127.8 

43.3 

20 

110 

69.9 

35.4 

10 

44 

34.9 

20.4 

The  steel  was  included  by  adding  its  concrete  equivalent. 

Fig.  116  illustrates  a  standard  form  of  abutment  used  by  the 
Wabash  R.R.  Co.* 

193.  Retaining-walls  Supported  at  the  Top. — Frequently  a 
retaining-wall  may  be  supported  at  the  top.  In  such  a  case  it  is 
designed  as  a  simple  beam  supported  at  the  top 
and  bottom;  or  vertical  ribs  or  beams  may  thus 
be  calculated  and  the  slab  reinforced  horizontally 
and  supported  by  these  ribs. 

A  wall  AB  (Fig.  117)  acted  upon  by  a  pressure 
uniformly  varying  from  zero  at  the  top  to  a  maxi- 
mum at  the  bottom  will  be  subjected  to  a  bending 
moment  whose  maximum  value  will  be  determined. 
Let  the  pressure  be  that  due  to  a  fluid  weighing 
p  Ibs./f t3.  Then  P  =  \j>W,  RI  =  $P  =  %ph2,  R2  -  £M2- 

The  bending  moment  M  at  a  distance  x  below 
A  =  Rix  -  Ipx*  -  p /6  (h*x  -  x3) . 

This  is  a  maximum  for  x  =  h\'  %  =  .58h.    The  maximum  moment 
is  then 

M  =  .064p#».     .    ...    .    .    .     (8) 

If  the  pressure  is  water  pressure,  as  in  a  reservoir,  the  value 
of  the  maximum  moment  becomes  equal  to 

M  =  4h3,     .  - (9) 

where  the  units  are  the  foot  and  pound.    For  an  earth  retaining- 
wall  with  p  =  20,  then  Af  =  1.3A3,  etc. 

*  Ry.  Rev.,  Vol.  XLV,  1905,  p.  523. 


FIG.  117. 


§194] 


DAMS. 


383 


DAMS. 

194.  The  dam  is  a  form  of  retaining-wall,  but  is  subject  to 
somewhat  different  conditions  as  to  pressures.  For  this  case  a 
form  of  wall  as  shown  in  Fig.  118  is  poorly  adapted,  owing  to 
the  fact  that  the  water  pressure  will  probably  penetrate  beneath 
the  floor  DC  and  exert  an  upward  force  nearly  equal  to  the 
downward  pressure,  thus  destroying  the  usefulness  of  the  floor 
EC.  To  obviate  these  objections  the  wall  AE  must  be  brought 
back  to  the  point  C.  Increased  stability  will  then  be  secured 
by  making  it  inclined.  In  this  position  it  will  naturally  be 
supported  by  transverse  walls  or  buttresses,  resting  on  a  floor 
DC,  or  directly  on  the  foundation  material,  as  shown  in  Fig. 
119.  The  water  pressure  on  the  floor  may  then  be  relieved  by 


D          E  C 

FIG.  118.      - 


FIG.  119. 


drain-openings  allowing  free  exit  for  seepage-water.  Thus 
built  it  forms  a  stable  and  efficient  type  of  dam.  Its  design  as 
to  stresses  and  sections  is  simple  and  obvious.  The  wall  or 
floor  AC  may  be  supported  directly  on  the  cross-walls  and  re- 
inforced with  longitudinal  rods,  or  longitudinal  beams  may  be 
used  as  shown  and  the  slab  supported  on  these.  The  pressure 
on  the  foundation  is  determined  by  considering  the  resultant 
of  water  pressure  and  weight  of  dam.  The  buttresses  or  cross- 
walls  are  subjected  only  to  compressive  stresses.  Ample  longi- 
tudinal reinforcement  should  be  provided  to  thoroughly  bind 
the  structure  together.  Dams  are  often  subjected  to  dynamic 
loads  as  well  as  static  pressures,  and  sections  must  be  provided 
more  liberally  than  in  many  other  structures. 


384 


RETATNING-WALLS   AND  DAMS. 


[Cn.  IX 


The  form  shown  in  Fig.  119  is  not  suited  to  act  as  a  spillway 
except  for  low  falls.  For  a  spillway  the  down-stream  edge  of 
the  buttresses  is  also  covered  with  a  floor  which  may  be  curved 
in  the  usual  manner. 


Crew 


134.00 


FIG.  120.— Dam  at  Schuylerville,  N.  Y. 

Fig.  120  illustrates  a  dam  of  this  type  built  at  Schuylerville, 
N.  Y.,  by  the  Ambursen  Hydraulic  Construction  Co.*  A  foot- 
way is  provided  for  in  the  interior.  The  design  as  to  strength 
is  obvious. 


*  Eng.  News,  Vol.  LIII,  p.  448. 


CHAPTER  X. 

MISCELLANEOUS     STRUCTUEES. 
GIRDER   BRIDGES   AND   CULVERTS. 

195.  For  short  spans,  the  girder  bridge  or  box  culvert  is 
likely  to  be  a  more  economical  form  than  the  arch,  owing  to 
the  less  rigid  requirements  for  foundations  and  abutments. 
For  purposes  of  analysis  this  type  of  structure  may  be  divided 
roughly  into  three  classes:  (1)  Simple  spans  in  which  the 
girder  rests  upon  independent  abutments  or  piers;  (2)  con- 
crete trestles  or  bridges  in  which  the  girders,  abutments  and 
piers  form  a  monolithic  structure;  and  (3)  pipe  culverts  and 
box  culverts  built  as  square  or  rectangular  pipes. 

196.  The  Simple  Beam  Bridge.  —  These  are  designed  in  the 
same  manner  as  any  other  concrete  floor.  Spans  up  to  20  to 
30  feet  may  well  be  made  as  a  simple  slab  or  uniform  thickness 
spanning  the  opening.  For  railroad  structures  the  loads  are 
relatively  so  large  that  shearing  stresses  will  usually  require 
careful  attention.  For  longer  spans  a  gain  in  economy  will 
result  by  the  use  of  main  horizontal  girders  of  relatively  great 
depth,  with  a  floor  supported  by  the  girders  and  reinforced 
transversely.  The  bridge  may  be  made  either  a  "  through  "  or 
"  deck  "  girder,  according  to  the  requirements  of  the  case,  the 
latter  being  the  more  economical.  Floors  of  reinforced  con- 
crete are  also  used  for  steel  truss  and  girder  bridges  to  a  con- 
siderable extent  where  a  solid  floor  is  desired.  The  details  are 
arranged  in  a  variety  of  ways,  but  the  calculation  and  design 
of  the  reinforcement  to  meet  the  given  conditions  require  no 
special  consideration.  The  proper  allowance  for  impact  is  an 

385 


386  MISCELLANEOUS    STRUCTURES.  [Cn.  X. 

important  point  in  this  connection.  Durability  is  an  important 
factor  favorable  to  the  use  of  reinforced  concrete  for  bridge 
floors. 

197.  Concrete  Trestles.  —  Where  several  short  spans  are 
required  and  concrete  is  used  for  both  the  girders  and  the 
piers,  the  latter  may  usually  be  made  of  comparatively  small 
cross-section,  —  much  smaller  than  possible  if  ordinary  masonry 
be  used.  The  structure  then  approaches  the  ordinary  floor 
and  column  construction  in  the  relations  of  its  parts.  The 
piers,  if  lightly  loaded,  may  consist  merely  of  two  or  more 
columns  connected  by  a  suitable  portal.  In  some  extreme 
cases  designs  have  been  carried  out  in  which  the  supporting 
piers  or  towers  have  been  arranged  in  a  manner  similar  to  a 
steel  trestle,  even  to  the  diagonal  bracing.  It  would  seem, 
however,  that  the  treatment  of  concrete  should  be  on  some- 
what different  lines  than  is  best  suited  to  such  a  material  as 
steel,  and  that  structural  forms  in  concrete  should  be  somewhat 
massive  and  limited  in  general  to  the  beam  and  the  compression 
member. 

Where  the  piers  are  made  small,  as  here  assumed,  they  must 
be  built  rigidly  in  connection  with  the  girders  of  one  or  more 
spans,  as  are  the  columns  in  a  building.  The  girders  must  be 
designed  with  proper  reference  to  their  continuity,  and  the  piers 
must  be  able  to  resist  a  certain  amount  of  bending  moment. 
This  moment  can  be  estimated  in  the  manner  suggested  in 
Chapter  VII,  Art.  167. 

As  an  example,  let  Fig.  121  represent  a  concrete  trestle  of 
monolithic  construction.  The  girders  are  continuous  and  the 


FIG.  121. 


piers  are  rigidly  attached  to  them.     The  greatest  moment  in 
the  pier  BF  will  occur  when  one  of  the  spans  AB  or  BC  is  loaded. 


§  198.] 


PIPE    AND    BOX    CULVERTS. 


387 


Suppose  BC  be  loaded.  Then  calculate  the  negative  moment 
at  B,  assuming  BC  to  be  fixed  at  the  ends.  This  moment  will 
be  equal  to  —  iV  pi2,  where  p  =  load  per  foot  and  I  =  span 
length.  Now  this  moment  is  distributed  at  the  joint  B  among 
the  three  members  A  Bf  BF,  and  BC  in  proportion  to  the 
value  of  I/I  for  the  three  members,  the  length  I  being  taken  as 
the  estimated  length  to  the  point  of  inflection  in  each  case 
(the  full  length  of  BF).  This  will  determine  approximately 
the  moment  in  BF.  The  maximum  negative  moment  in  BC 
and  A  B  will  occur  when  both  spans  are  loaded  and  will  be 
approximately  equal  to  TV  pF.  (See  Chapter  VII,  Art.  155.) 
The  end  piers  or  abutments  must  be  designed  also  as  retaining 
walls. 

198.  Pipe  and  Box  Culverts.  —  For  small  openings  the  mono- 
lithic pipe  or  box  form  is  very  advantageous.     This  form  of 
structure  is  a  complete  opening  in  itself  and  so  long  as  intact 
will  do   good   service.    Considerable    settlement,   as   a  whole, 
may  be  permissible,  and  hence  solid  foundations  may  not  be 
needed. 

The  cross-section  may  be  circular,  elliptical  or  rectangular. 
Theoretically,  the  elliptical  form  is  the  best  as  corresponding 
more  nearly  to  the  requirements  for  resisting  the  earth  pressure. 
The  circular  is  practically  as  good  for  small  openings,  while 
for  large  openings  the  rectangular  form  will  often  be  the  best 
on  account  of  its  simplicity  and  the  lesser  head  room  required. 
Where  the  culvert  is  manufactured  at  a  shop  and  transported 
to  the  site,  the  circular  or  elliptical  forms  will  usually  be  the 
most  advantageous.  As  the  loads  coming  upon  such  structures 
are  not  accurately  known  an  exact  analysis  of  the  stresses  is 
impossible,  but  the  results  obtained  for  certain  simple  cases 
will  be  useful  as  a  guide  to  the  judgment.  The  general  method 
of  analysis  employed  in  Chapter  VIII  has  been  used.  The 
details  of  the  analysis  will  be  omitted. 

199.  The     Circular    Culvert.  —  Two     cases     have     been 
analyzed;   (1)  for  a  uniform  load,  and  (2)  for  a  concentrated 
load. 


388  MISCELLANEOUS    STRUCTURES.  [Cn.  X. 

Case  I;  Uniform  load.  (Fig.  122.)  It  is  assumed  that  the 
pressure  on  the  pipe  is  exerted  in  parallel  lines  (as  downward 
and  upward)  and  is  uniformly  distributed  with 
respect  to  a  plane  perpendicular  to  the  direction 
of  the  pressure. 

/'"a    "X^ 

Let  d  =  diameter  of  pipe; 

p  =  pressure  per  unit  area  as  measured  per- 

pendicularly to  the  pressure;  \S         J 

M  =  bending  moment  in  pipe  in  a  length  of     1  1  H  M  1  1  1 

one  unit;  v__^—  i* 

~ 


s        >v 

(A 


Then  the  following  equations  result. 


FlG> 


Ma  =  Mb  =  TV  pd2     .......      (1) 

......     (2) 


If  the  lateral  pressure,  measured  in  a  similar  way,  be  p'  per 
unit  area,  then  the  moments  due  to  this  pressure  will  be 


*    ......     (3) 

and  Mc  =  Md  -  TV  p'd2    .......     (4) 

For  equal  horizontal  and  vertical  forces  (equivalent  to  a  uni- 
form radial  pressure),  the  moments  at  all  points  are  zero. 
Usually  the  lateral  pressure  will  be  much  less  than  the  vertical 
pressure;  probably  not  more  than  one-fourth  or  one-fifth  as 
much.  Assuming  a  ratio  of  one-fourth,  the  resulting  total 
bending  moments  at  the  points  a,  b,  c,  d,  will  be  A  pd  2, 

positive  at  the  top  and  bottom  and  negative  at 

the  sides. 

Case  II;  Concentrated  loads  at  opposite  points 

(Fig.  123). 
In  this  case  the  moments  are 

Ma  =  Mb  =  .16  Pd      ......     (5) 

FIG.  123.  Mc  =  Md=-.09Pd      .....     (6) 


§200.] 


PIPE  AND  BOX    CULVERTS. 


389 


200.      The   Rectangular   Culvert.  —  Case   I;    Uniform    loads 
(Fig.  124). 

Let  Zt  =  width  of  culvert; 
12  =  height  of  culvert; 

7t  =  moment  of  inertia  of  top  and  bottom,  assumed  as  equal ; 
72  =  moment  of  inertia  of  sides; 
p  =  vertical  load  and  foundation  reaction  per  unit  area. 


Then 


pi 


8 


l./I,  +  12/I2 
Me  =  M<  =  Ma-lpl*      .     . 

The  moments  at  e  and  /  are  equal  to  M  c. 


.    .    .    .     (7) 
(8) 


J. 

1 
f 

1 

h—  3 

Te 

JL 

t 

a 

c                                    d 
b 

7 

tttttttntttttt 


FIG.  124. 

For  a  square  culvert  with  uniform  section  Mt 


pP  and 


For  equal  vertical  and  lateral  loads  the  moments  in  the  square 
culvert  become  Ma  =  Mc  =  +  &  pi2  and  Me  =  -  ^  pi2  as  in 
a  beam  with  fixed  ends. 

Case  II;  Concentrated  loads.     (Fig.  125.) 

For  vertical  loads  applied  centrally, 


__pit 

M   -  M-  —  . 


™ 

(9, 


4       /1//1  +  12/I, 
Me  =  Md  =  Ma-\Pl,        ......    .     (10) 


390  MISCELLANEOUS    STRUCTURES.  [Cfc.  X. 

For  the  square  form,  Ma  =  tfc  Pl^  and  Mc  =  -  TV  P^  ;  and 
for  equal  lateral  and  vertical  forces  Ma  =  Mc  =  -|  Pl^  and 
Af e  =  —  \  Pli  as  for  fixed  beams. 

201.  Arrangement  of  Reinforcement. — The  bending  moments 
here  determined  are  based  on  the  assumption  that  the  entire 
section  is  reinforced  so  as  to  act  as  a  monolithic  structure. 
This  of  course  requires  proper  reinforcement  for  negative  as 
well  as  positive  moments. 

In  the  circular  form  a  wire  mesh  is  convenient,  especially 
for  small  diameters.  A  single  mesh  will  be  sufficient,  placed 
near  the  intrados  at  top  and  bottom  and  near  the  extrados 
at  the  sides,  crossing  the  central  axis  at  about  the  quarter 
point. 

In  the  rectangular  form,  if  reinforcement  for  negative  moments 
at  the  corners  is  omitted,  then  the  four  sides  will  act  as  simple 
beams,  the  concrete  cracking  more  or  less  on  the  outside  near 
the  corners. 

Longitudinal  reinforcement  should  be  provided  to  some 
extent.  Where  foundations  are  good  a  very  small  amount  will 
be  sufficient,  but  if  settlement  is  likely  to  occur  the  longitudinal 
reinforcement  becomes  of  much  importance.  The  entire  cul- 
vert will  act  as  a  beam  subjected  in  the  main  to  positive  bend- 
ing moments.  Most  of  the  reinforcement  should  therefore  be 
placed  along  the  bottom  of  the  culvert. 

20  ia.  Tests  of  Reinforced  Concrete  Rings  and  Culvert 
Pipe. — Large  reinforced  concrete  rings  and  pipe  have  been  tested 
by  Professor  Talbot,  with  results  agreeing  closely  with  the 
theoretical  analysis  of  Art.  199.  Loads  were  applied  in  two 
ways,  (a)  as  concentrated  loads,  in  which  the  pipe  was  sup- 
ported along  an  element  at  the  bottom  and  the  load  was  applied 
along  an  element  at  the  top;  and  (6)  as  distributed  loads,  in 
which  the  pressures  at  bottom  and  top  were  distributed  as 
uniformly  as  possible  over  the  entire  horizontal  projection  of 
the  pipe  by  means  of  a  carefully  constructed  sand  box.*  These 

*  For  full  details  see  Bulletin  No.  22,  Eng.  Exp.  Sta.,  University  of 
Illinois,  1908. 


§  201a.] 


PIPE    AND    BOX    CULVERTS. 


391 


methods  of  loading  correspond  to  the  cases  of  concentrated 
and  distributed  loads  discussed  in  Art.  199.  All  rings  and  pipe 
were  48-in.  internal  diameter  and  4-in.  thick,  the  rings  being 
24  in.  long,  and  the  pipe  sections  102-104  in.  long.  The  latter 
were  made  with  the  usual  bell  end.  The  reinforcement  con- 
sisted of  J-in.  corrugated  bars  except  in  No.  982,  in  which 
J-in.  bars  were  used,  and  in  No.  988,  in  which  No.  3  Clinton 
wire  mesh  was  used.  The  rings  were  made  circular  and  the 
reinforcement  placed  near  the  intrados  at  top  and  bottom  and 
near  the  extrados  at  the  sides.  In  the  pipe,  the  reinforce- 
ment was  made  circular  and  the  concrete  cast  with  a  vertical 
diameter  4  ir.  greater  than  the  horizontal,  giving  a  similar 
relative  position  for  the  reinforcement.  The  results  are  given 
in  Tables  23  and  24.  In  these  tables  the  value  of  t  is  the 
net  or  effective  thickness  of  the  pipe  as  measured  to  the  center 
of  the  steel.  In  the  last  column  of  Table  23  the  theoretical 
strength  is  the  strength  calculated  by  means  of  eq.  (5),  Art.  199, 
assuming  the  bending  resistance  of  the  pipe  to  be  .87  Afst  per 
lineal  foot,  using  the  yield-point  of  the  steel  for  the  value 


TABLE  No.  23. 

RESULTS  OF  TESTS  ON  REINFORCED  CONCRETE  RINGS. 

(TALBOT.) 

(Concentrated  Loads.) 
Diam.  of  rings  =  48  ins.;  thickness  =  4  ins.;  age  1-3  mos. 


Reinforce- 

Load at 

Maximum 

Ratio  of 

No. 

ment, 

First  Crack, 

Load, 

t  inches. 

Theoretical 

Per  Cent. 

Lbs/lin.ft. 

Lbs/lin.ft. 

to  Actual 

Strength. 

926 

0.73 

1500 

2850 

2.75 

1.12 

928 

0.80 

1400 

3550 

2.5 

0.82 

931 

0.73 

2150 

2500 

2.75 

1.28 

932 

0.66 

1500 

3000 

3.0 

1.16 

933 

1.00 

1170 

3170 

2.0 

0.73 

934 

0.80 

1300 

3150 

2.5 

0.92 

952 

1.00 

1000 

2350 

2.0 

0.99 

953 

0.89 

1200 

3600 

2.25 

0.73 

971 

0.73 

1500 

4120 

2.75 

0.77 

392 


MISCELLANEOUS    STRUCTURES. 


[Cn.  X. 


TABLE  No.  24. 

RESULT  OF  TESTS  ON  REINFORCED  CONCRETE  RINGS  AND 

PIPES. 

(TALBOT.) 
(Distributed  Loads.) 

Diam.  of  rings  and  pipe  =  48  ins.;  thickness  =  4  ins.;  age  of  rings  1-3  mos.; 
age  of  pipe  4-6  months. 

REINFORCED  CONCRETE  RINGS. 


No. 

Reinforce- 
ment, Per 
Cent. 

Load  at 
First  Crack. 
Lbs/lin.ft. 

Critical 
Load, 
Lbs/lin.ft. 

Maximum 
Load, 
Lbs/lin.ft. 

t  inches. 

Ratio  of 
Theoretical 
to  Actual 
Strength  at 
Critical 
Load. 

923* 
921 
922 
927 
951 
972 
976 
977 

0.80 

0.80 

0.80 
0.80 
0.80 
0.73 
0.66 
0.66 

2250 
3500 
3250 
3250 
3200 
4500 
4000 
4000 

7000 
10000 

10000 
8000 
9000 
8000 
9000 
10000 

10500 
23500 
18500 
26000 
25000 
17500 
19000 
21000 

2.5 
2.5 
2.5 
2.5 
2.5 
2.75 
3.0 
3.0 

1.06 

0.74 
0.74 
0.93 
0.83 
1.03 
0.99 
0.89 

REINFORCED  CONCRETE  PIPE. 


981 

0.66 

8360 

19500 

31500 

3.0 

0.55 

982 

1.39 

10960 

15000 

24800 

3.0 

1.49 

983 

0.66 

4950 

12500 

23800 

3.0 

0.86 

988 

0.88 

6700 

9000 

31400 

3.0 

*  No  lateral  restraint. 

of  /*.  This  was  taken  at  46,400  lbs/in2  for  the  rings  and 
55,000  lbs/in2  for  the  pipe.  The  tests  showed  that  the  primary 
cause  of  failure  was  the  failure  of  the  steel  in  tension.  In 
Table  24,  similarly,  the  theoretical  strength  is  determined  on 
the  basis  of  eq.  (1)  of  Art.  199.  In  the  case  of  the  distributed 
load  tests  the  "  critical  load  "  was  estimated  as  the  load 
beyond  which  the  increased  resistance  was  primarily  due  to 
increased  lateral  resistance  of  the  sand  filling  and  not  of  the 
ring  or  pipe  itself.  The  ultimate  resistance  would  obviously 
be  chiefly  dependent  upon  the  character  of  the  filling  about 
the  pipe. 


§  201a.] 


PIPE   AND    BOX    CULVERTS. 


393 


394 


MISCELLANEOUS   STRUCTURES. 


[Cn.  X. 


The  results  of  the  concentrated  load  tests  show  close  agree- 
ment between  the  theoretical  and  experimental  values.  This 
means  merely  that  the  resisting  moment  of  a  pipe  may  be 
taken  the  same  as  that  of  a  straight  beam  of  the  same  thickness 
and  reinforcement.  The  tests  under  distributed  loads  show 
that  the  theoretical  strength  can  be  secured  by  very  careful 
bedding  and  testing.  It  would  seem  that  in  practice  nothing 
should  be  allowed  for  lateral  support  and  that  the  theoretical 
moment  of  1/16  pd2  may  be  used  if  the  filling  and  beddug 
are  carefully  done.  If  poorly  done  a  larger  bending  moment 
will  exist  and  will  need  to  be  considered. 

202.  Illustrative  Examples. — Fig.  126  illustrates  a  simple 
beam  bridge  or  "  trestle'0  on  the  Chicago,  Burlington  and 
Quincy  R.R.*  The  girder  consists  of  a  slab  twenty-four  inches 
in  thickness,  reinforced  as  shown  in  the  illustration.  The  piers 
are  separate  structures. 

Fig.  127  represents  a  concrete  highway  bridge  as  an  over- 
head crossing  of  the  Big  Four  R.R.  This  design  illustrates 
the  deep  girder  with  floor-slab  reinforced  transversely,  and  also 
the  "  trestle"  in  which  the  piers  are  columns  built  as  one  piece 
with  the  girders. | 


CROSS  SECTION 


FIG.  128. 


Fig.  128  illustrates  a  standard  design  for  a  monolithic  box 
culvert.     It  is  not   reinforced  for  negative   moment  at   the 


*  R.  R.  Gaz.,  Vol.  XL,  1906,  p.  713. 
t  R.  R.  Gaz.,  Vol.  XL,  1906,  p.  497. 


203.] 


CONDUITS    AND    PIPE   LINES. 


395 


corners.  This  form  of  construction  is  applicable  to  many  other 
structures  as  subways,  tunnel  linings,  etc.  No  special  con- 
sideration of  these  various  applications  of  the  reinforced  beam 
is  required  in  this  place.  A  clear  understanding  of  the  general 
principles  of  reinforced  concrete  design  will  enable  the  details 
to  be  suitably  modified  to  meet  the  conditions  of  the  case. 


CONDUITS  AND  PIPE  LINES. 

203.   For  conduits  not  under  pressure,  large  sewers  and  the 
like,  reinforced  concrete  lends  itself  to  convenient  and  economi- 


•3"  #  10  Exp.  Metal 


FIG.  129. 


cal  construction.  As  to  the  analysis  and  design,  these  struc- 
tures are  only  special  cases  of  the  monolithic  pipe  or  box 
discussed  in  preceding  articles.  The  character  of  the  foundation 


Twisted  Steel  Rods  l'.apart 


LappedJL' 
and  Wired 


*10  Expanded  Metal 


FIG    130. 


396  MISCELLANEOUS    STRUCTURES.  [Cn.  X. 

and  convenience  in  construction  will  lead  to  various  modifi- 
cations of  design. 

Fig.  129  is  a  typical  cross-section  of  a  large  sewer  for  Harris- 
burg,  Pa.  A  mesh  of  expanded  metal  is  used  for  reinforce- 
ment, arranged  to  resist  positive  moments  excepting  at  bottom 
and  corners. 

Fig.  130  illustrates  a  large  conduit  of  the  Jersey  City  Water 
Supply.  This  section  is  employed  where  the  bottom  is  soft, 
special  reinforcement  being  used  in  the  invert.  The  position 
of  the  reinforcement  to  carry  positive  moments  at  crown  and 
negative  moments  at  sides  should  be  noted. 

Reinforced  concrete  has  also  been  used  to  some  extent  for 
pipes  under  pressure,  but  it  is  very  difficult  to  secure  impervious- 
ness  under  heads  of  considerable  magnitude.  In  pressure 
pipes  the  tensile  stress  is  entirely  taken  by  the  steel,  the  concrete 
furnishing  merely  the  impervious  layer  and  resisting  bending 
due  to  earth  loading. 

TANKS,  RESERVOIRS,  BINS,  ETC. 

204.  For  covered  reservoirs  reinforced  concrete  is  very  well 
adapted.  The  rectangular  form  with  flat  cover  is  usually  the 
most  convenient;  its  design  involves  the  same  features  as  build- 
ing design  with  the  additional  one  of  imperviousness.  Elevated 
towers  and  tanks  may  also  be  made  of  concrete,  but  high 
pressures  are  difficult  to  deal  with. 

Bins  and  coal  pockets  are  structures  for  which  concrete  is 
well  adapted.  For  the  storage  of  coal  unprotected  steel  is  not 
durable,  but  reinforced  concrete  furnishes  an  almost  ideal 
material,  lending  itself  readily  to  the  necessary  form  for  strength 
and  furnishing  the  desired  durability. 

Reinforced  concrete  is  advantageously  used  in  other  minor 
forms  of  structures  and  structural  elements.  Noteworthy 
among  such  uses  are  its  employment  for  piles,  fence  posts,  and 


§204.]  TANKS,  RESERVOIRS,  BINS,  ETC.  397 

poles  for  various  purposes.  For  piles  it  is  especially  advantage- 
ous in  situations  where  continual  submergence  is  not  certain. 
For  further  illustrations  the  reader  is  referred  to  the  larger 
works  on  the  subject  and  especially  to  the  American  works  of 
Messrs.  Buell  and  Hill  and  Mr.  Reid. 


CHAPTER  XI. 

REINFORCED-CONCRETE  CHIMNEYS. 

205.  General  Description. — Over  400  reinforced-concrete 
chimneys  have  already  been  built  in  America,  one  at  least 
350  ft.  high,  and  18  ft.  in  diameter.  Fig.  131 
represents  the  general  features  of  such 
chimneys  as  usually  built;  they  are  the 
shaft  or  outer  shell,  the  base,  and  the  lining 
or  inner  shell. 

The  outer  shell  is  usually  made  of  two 
or  more  thicknesses,  the  maximum  thick- 
ness being  about  12  in.  in  a  large,  tall 
chimney,  and  the  minimum  is  as  small  as 
4  in.  in  some.  Outer  shells  are  usually  built 
with  an  offset  at  the  top  of  the  lining  to 
avoid  a  large  change  in  internal  diameter, 
and  for  architectural  effect,  but  as  an  offset 
:'s  a  weak  place  unless  very  well  reinforced, 
some  chimneys  have  been  built  with  uni- 
form outer  diameter.  Shells  are  generally 
made  cylindrical,  but  at  least  one  firm  of 
builders  advertises  taper  or  conical  chim- 
neys. 

Bases  are  generally  square  or  octagonal 
in  plan,  and  thinner  at  the  perimeter 
than  at  the  shells;  but  small  bases  have 
been  built  uniform  in  thickness  for  sim- 
plicity, and  large  ones  have  been  built 
thin  near  the  center  as  well  as  at  the 
perimeter  to  effect  a  saving  of  con- 

FIG.  131.  crete. 

Linings  for  the  protection  of  the  outer  shell  from  excessive 
heat,  were  built  to  the  top  in  some  of  the  earlier  chimneys. 

398 


§  206.]  DESIGN    OF    A   CHIMNEY.  399 

Now  they  are  generally  built  to  about  one-third  the  height; 
but  there  are  some  ch  mneys  in  use  without  any  1  ning.  There 
s  no  rigid  connection  between  the  shells  except  at  the  base, 
hence  the  lining  is  free  to  expand  and  contract  and  is  not 
subjected  to  wind  stresses.  The  thickness  is  generally  4  or 
5  in.  and  the  air  space  between  the  shells  is  made  as  small 
as  the  forming  will  allow,  about  4  in.  The  air  space  com- 
municates with  the  outer  air  through  vent  holes. 

206.  Design  of  a  Chimney. — The  inner  diameter  and  the 
height  of  the  chimney  are  determined  by  the  requirements  of 
draft  and  capacity — matters  which  do  not  fall  within  the 
scope  of  this  work. 

The  outer  shell  may  be  made  4  to  6  in.  at  the  top,  depend- 
ing on  the  diameter,  the  thickness  to  be  increased  one  or  more 
times  in  the  height  as  required.  While  many  changes  in 
thickness  would  result  in  a  saving  of  concrete,  it  must  be 
noted  that  each  change  means  an  alteration  in  forms  and 
hence  an  expense.  Thicknesses  are  chosen  tentatively,  also 
amounts  of  reinforcements,  both  vertical  and  circumferential, 
and  then  various  sections  are  investigated  for  stress.  Methods 
for  computing  wind  and  temperature  stresses  are  explained  in 
Arts.  207-209  and  211-214. 

A  base  should  be  made  with  such  an  extent  of  bottom 
that  its  greatest  pressure  on  the  earth  due  to  weight  of  chim- 
ney, weight  of  earth  filling  over  the  base,  and  wind  pressure, 
will  not  exceed  the  permissible  limit;  and  the  base  itself  must 
be  strong  enough  to  withstand  the  pressures  on  its  top  and 
bottom.  Methods  for  investigating  these  points  for  a  given 
base  are  explained  in  Arts.  215-217. 

Linings  are  designed,  as  yet,  by  precedent.  In  the  report 
on  his  investigations  of  reinforced-concrete  chimneys,*  Sanford 
E.  Thompson  recommends  that  for  temperatures  above  750°  F. 
fire-brick  be  used  for  linings,  but  for  lower,  suitable  cement 
mortar  may  be  safely  used.  Inasmuch  as  a  number  of  outer 

*  For  abstract,  see  Eng.  News,  Jan.  9,  1908. 


400  REINFORCED-CONCRETE    CHIMNEYS.  [Cn.  XL 

shells  have  cracked  badly  near  the  top  of  the  lining,  he  sug- 
gests that  linings  be  built  higher  than  one-third  the  height 
or  else  that  the  outer  shell  be  extra-reinforced  near  the  top 
of  the  lining.  No  reports  of  injured  linings  were  received 
by  him.  The  usual  thickness,  4  in.  for  moderate  heights, 
and  the  reinforcements  used  would  seem  adequate  but  it 
should  be  noted  that  comparatively  few  owners  have  examined 
the  linings  of  their  chimneys.  Published  descriptions  of 
reinforcements  are  meager.  In  one  lining,  90  ft.  high,  the 
percentage  of  hoop  steel  is  J  and  the  hoops  are  spaced  20  in. ; 
in  another  60  ft.  high,  it  is  J%  and  the  spacing  is  36  in.  In 
the  latter  the  percentage  of  vertical  steel  is  1.8. 

207.  Wind  Stresses  in  the  Outer  Shell. — On  a  horizontal 
section  of  a  chimney  sustaining  no  wind  pressure,  the  "  fiber 
stress "  in  the  concrete  is  a  uniform  compression.  Wind 
pressure  changes  this  uniform  stress,  increasing  the  inten- 
sity of  the  compression  on  the  lee  side  and  decreasing  it  on 
the  windward.  The  decrease  may  be  larger  than  the  pre- 
existent  intensity,  the  net  result  being  a  tensile  stress.  Two 
cases  will  be  distinguished;  in  both  it  is  assumed,  just  as  in 
the  most  widely  used  flexure  formulas  for  the  working  strength 
of  an  ordinary  reinforced-concrete  beam,  that  the  fiber  stress 
is  a  uniformly  varying  one. 

Notation. — In  this  connection  see  Figs.  132  to  136.    Also  let 

A  =  area  of  chimney  section  under  consideration; 

AS= total  area  of  all  steel  sections  there; 

W  =  weight  of  superincumbent  portion  of  chimney; 

P  =  wind  pressure  on  that  portion; 

M  =  bending  moment  at  the  section; 

e  =  distance  from  the  center  of  the  section  to  where  the 
resultant  of  the  weight  and  wind  pressure  cuts  the 
section,  "  eccentric  distance"; 

x>/c  =  unit  stress  in  concrete  adjacent  to  the  steel  at  lee  side; 
x~/c/  =  unit  stress  in  concrete  adjacent  to  steel  at  windward  side; 
/=unit  stress  on  concrete  at  the  lee  side; 

/'=unit  stress  on  concrete  at  the  windward  side; 


§  208.]  WIND    STRESSES.  401 

/s  =  unit  stress  on  steel  at  the  windward  side; 
ra  =  a  coefficient  such  that  fc=mW/A; 
w'  =  a  coefficient  such  that  /«•'  =  m'W/A ; 
p  =  steel  ratio,  i.e.,  AS/A;  and 

n= ratio  of  modulus  of  elasticity  of  steel  to  that  of  concrete 
(taken  as  15  in  all  numerical  work  following). 

208. — Case  I.  The  stress  at  the  windward  side  is  compressive 
or  a  tension  of  low  intensity,  say,  50  lbs/in.2  This  case  obtains 
in  all  sections  where  the  resultant  of  W  and  P  falls  within 
or  not  far  without  the  kern*  of  the  section.  (The  kern  is  a 
circle  concentric  with  the  hollow  circle,  its  radius  being 
Jr2[l  +  (r1/r2)2].  Since  in  chimneys  r\/r<z  is  nearly  1,  the 
kern  radius  is  nearly  Jr2;  it  may  be  taken  as  Jr;).  Fig.  133a 
represents  the  variation  in  the  concrete  stress  (wholly  com- 
pressive) when  the  resultant  falls  well  within  the  kern,  and 
Fig.  1336  represents  it  (part  tensile)  when  the  resultant  falls 
outside. 

First  Method. — This  is  carried  out  graphically  by  means 
of  a  diagram  (Fig.  132);  it  will  be  described  by  examples, 
and  then  the  analysis  on  which  the  diagram  is  based  will 
be  given.  At  the  base  of  the  diagram  there  are  given  values 
of  the  eccentricity,  e/r  from  0  to  0.8;  at  the  left  side  values  of 
the  coefficient  m,  and  at  the  right  values  of  the  coefficient  mr. 


*  Imagine  all  the  forces  acting  on  either  side  of  a  section  of  a  beam,  or 
column,  etc.,  to  be  compounded  into  a  resultant  R,  or  if  that  is  impossible, 
into  two  forces:  one,  N,  perpendicular  to  the  section  and  one  in  the  sec- 
tion (always  possible).  If  R  or  N  cuts  the  section  at  its  centroid,  then 
the  normal  stress  at  the  section  is  all  one  kind,  tension  or  compression, 
and  it  is  a  uniform  stress.  If  R  or  N  cuts  the  section  and  not  very  eccen- 
trically, then  the  normal  stress  will  still  be  of  one  kind  but  it  will  not  be 
a  uniform  one.  That  part  of  the  cross-section  within  which  R  or  N  must 
cut  the  section  in  order  that  the  normal  stress  may  be  of  one  kind,  is  called 
the  kern  of  the  section. 

When  the  normal  stress  varies  uniformly,  then  the  kern  for  any  section 
can  be  determined  easily.  For  this  kind  of  normal  stress  the  kern  for  a 
hollow  circle  is  described  above  and  for  a  square  an  octagon  and  a  circle 
in  Art.  216. 


402 


REINFORCED-CONCRETE    CHIMNEYS.  [Cn.  XI. 


OJl  0.5  0.3  0.4  0.5  0.6  0.7 


0.8 

0 


05* 


0.1 


0.2  0.3  0.4  0.5  0.6 

Tallies  of  Eccentricity,  e--+r 

FIG.  132.— Wind  Stresses  in  Chimneys. 


0.7 


208.] 


WIND    STRESSES. 


403 


The  oblique  lines  relate  to  various  percentages  of  steel  from 
0  to  2,  and  may  be  called  "  percentage  lines."  It  will  be 
noticed  that  there  are  no  m'  percentage  lines  for  eccentricities 
less  than  0.5;  for  such  small  values,  the  stress  //  is  com- 


kr— * 


FIG.  133a. 


pressive  (and  less  than  fc)  and  there  is  no  need  ever  to  deter- 
mine these  small  values  of  //. 

Examples. — (1)  A  reinforced-concrete  chimney  is  150  ft.  high;  its 
outer  diameter  is  constant  and  equal  to  12  ft.  2  in.;  the  upper  100  ft. 
of  the  shell  is  6  in.  thick  and  the  lower  50  ft.  is  8  in.  thick.  At  a 
section  50  ft.  from  the  top  the  vertical  reinforcement  consists  of  sixteen 
J-in.  rods.  The  extreme  unit  stresses  at  this  section  are  required,  the 
chimney  being  under  wind  pressure  assumed  to  be  equivalent  to  30 
lbs/ft2  of  "projected  area." 

Taking  the  weight  of  concrete  as  150  lbs/ft3,  W  is  about  137,500  Ibs., 
and  since  M  is  about  5,475,000  in-lbs.,  e  =  M/W  =  39.8  in.,  and  e/r  =  0.57. 
Since  As  =  7.07  in2,  and  ,1  =  2636  in2,  p  =  0.0027  =  0.27%.  With  these 
values  of  p  and  e/r,  the  diagram  (Fig.  132)  gives  w  =  2.06  and  ra'  =  0.135; 
hence  /=2.06  TF/A  =  107  lbs/in2,  and  /'  =  0.135  W/A  =  7  lbs/in2.  This 
f  being  a  small  tension,  the  stress  condition  at  the  section  under  con- 
sideration does  fall  under  Case  I. 

(2)  At  the  section  100  ft.  below  the  top  of  the  chimney  tKe  vertical 
reinforcement  consists  of  forty-eight  f-in.  rods.  It  is  required  to  deter- 
mine the  extreme  unit  stresses  there. 

Here  W  is  about  275,000  Ibs.,  and  M  about  21,900,000  in-lbs.; 
hence  e  =  M/W  =  79.6  in.  and  e/r  =  1.137.  Since  As  =  21.2  in2,  and 
,4=2636  in2,  p  =  0.008  =  0.8%.  These  values  of  e/r,  and  p  fall  beyond 
the  limits  of  the  diagram ;  hence  the  stress  condition  does  not  fall  under 
Case  I  probably.  Substituting  in  eq.  (6),  it  will  be  found  that 
w'  =  1.15;  and  hence  /'  =  1.15  W/ A  =  120  lbs/in2  approximately;  this 
being  a  high  tension,  the  stress  condition  does  not  fall  under  Case  I  and 
the  methods  of  Case  II  should  be  applied  (see  ex.  1,  Art.  209). 


404  REINFORCED-CONCRETE  CHIMNEYS.  [Cn.  XI. 

Analysis  for  the  Diagram.  —  In  the  case  of  a  uniformly 
varying  stress,  the  average  unit  stress  for  any  portion  of  the 
section  equals  the  actual  unit  stress  at  the  centroid  of  that 
portion  (or  at  any  point  of  the  section  whose  distance  from 
the  neutral  axis  equals  that  of  the  centroid).  Hence  the 
average  unit  stress  in  the  concrete  is  i(/c+/c/);  also  the 
average  unit  stress  in  the  steel  is  %(fc+fc')n  (see  Fig.  133). 
(It  is  supposed  that  the  vertical  steel  is  securely  tied  to  the 
circumferential  so  that  the  former  will  not  buckle.) 

And  since  the  total  stress  on  the  section  equals  W, 

W.    .    .     .     .     (1) 


In  the  case  of  a  uniformly  varying  stress,  the  point  of  applica- 
tion of  the  resultant  of  the  stress  on  any  portion  of  the  section 
lies  at  a  distance  from  the  neutral  axis  equal  to  the  ratio 
between  the  square  of  the  radius  of  gyration  of  the  portion 
with  respect  to  the  neutral  axis  and  the  distance  of  the  centroid 
of  that  portion  from  the  same  axis.  Now  the  radius  of  gyra- 
tion of  the  concrete  section  is  nearly  the  same  as  that  of  the 
steel  circle  (radius  r),  and  hence  the  resultants  of  the  concrete 
and  steel  stresses  practically  coincide.  The  square  of  the 
radius  of  gyration  of  this  circle  with  respect  to  the  neutral 
axis  is  ir2+(l-&)2r2  (see  Fig.  133);  hence  the  arm  of  the 
resultants  with  respect  to  the  neutral  axis  is 


and  the  arm  with  respect  to  the   center  of  the  section  is 

r/2(l  —  k).     Since    the    sum  of    the    moments    of    these    two 

resultants    with    respect    to  the    center    equals    the    bending 
moment, 


From  eqs.  (1)  and  (2),  it  follows  that 

'-   ........  * 


§208.]  WIND    STRESSES.  405 

From  the  similar  triangles  in  Figs.  133a  and  1336,  and  eq.  (3), 
it  follows  that 

fc'=-fck/(2-k)=fc(l-2e/r)/(l  +  2e/r), 
and  if  this  value  be  substituted  in  (1),  then  the  equation  gives 


/(A\ 
-  i  i  „„  ~TI w 

hence,  also, 

fc'~'~  1+np  A' 
Since  by  definition,  fc  =  mW /A    and    /</  =  ' 

l  +  2e/r  l-2e/r 

m=^r- : ,    and    m  =^— .       ...     (6) 

1+np  1+np 

The  straight  line  in  Fig.  134  was  plotted  from  eq.  (3),  and 
all  lines  in  Fig.  132  from  eqs.  (6). 

It  should  be  noticed  that  fc  and  fc  are  not  the  unit  stresses 
for  the  extreme  fiber;  these  latter  might  be  obtained  from  the 
former  and  k  by  proportion  (see  Figs.  133a  and  1336),  or  by  the 

Second  Method. — This  is  the  ordinary  method  for  com- 
bining "  direct  "  and  flexural  stress;  it  gives  the  unit  stresses 
for  the  extreme  fibers.  Thus,  7  denoting  the  moment  of  inertia 
of  the  concrete-steel  section  about  a  diameter,  computed  as 
explained  below,  then 


.,     W    Mr2 
and  /"A-— ' 

If,  in  a  given  instance,  /'  comes  out  negative,  then  the  stress 
at  the  windward  side  of  the  section  under  consideration  is 
tensile.  The  greatest  compressive  unit  stress  in  the  steel  is 
less  than  nf,  and,  if  some  of  the  steel  is  under  tension,  its 


406 


REIN  FORCED-CONCRETE    CHIMNEYS. 


[Cn.  XL 


1.5 

1.4 
1.8 
1.2 

.1.1 
1.0 
0.9 
0.8 
0.7 

0.5 
0.4 
0.3 
0.2 
0.1 


/, 


Y 


alp  )TCfentagei 


0,5 


1.0  1.5 

Values  of  Eccentricity,  e~ 


2.0 


2.5 


FIG.  134. — Neutral  Axes  in  Chimneys. 


§209.] 


WIND    STRESSES. 


407 


greatest  tensile  unit  stress  is  less  than  nf.    The  first  of  these 
maxima  is  safe  if  /  is  safe,  and  the  second  is  insignificant. 

In  computing  the  moment  of  inertia  7,  the  steel  sections 
must  be  weighted  n-fold;  thus  let  7C  =  moment  of  inertia  of 
the  concrete  section  with  respect  to  a  diameter,  and  7S  =  moment 
of  inertia  of  all  the  steel  sections  with  respect  to  the  same 
line,  then  I=nL  +  Ic.  Instead  of  using  the  actual  steel  sec- 
tions to  compute  /,  one  may  substitute  with  sufficient  accuracy 
the  section  of  a  cylindrical  shell  rolled  from  the  steel,  the  mean 
radius  of  the  shell  being  r.  Then,  as  the  concrete  sectional 
area  is  practically  the  same  as  the  total, 


For  Ex.  (1),  7  =  i  2636(30  X  0.0027  X  702  +  672  +  732)  = 
6,731;685  in4;  hence  /=111  and  /'  =  -  7  lbs/in2,  the  negative 
sign  indicating  tensile  stress  at  the  windward  side.  For 
Ex.  (2),  7  =  6,970,000  in4,  /=  334  and  /'=  -126  lbs/in2. 

209.  Case  77.—  The  eccentricity  is  so  great  that  the  resultant 
stress  at  the  windward  side  is  a  tension  whose  intensity  is  so 
high  that  the  concrete  has  been  cracked  or  is  near  the  crack- 
ing stage;  in  other  words,  the  tensile  stress  condition  resem- 
bles somewhat  that  at  the  section  of  maximum  moment  in  an 
ordinary  reinforced-concrete  beam  under  full  safe  load.  In 
the  computation  for  this  case  the  tensile  strength  of  the 
concrete  will  be  entirely  neglected,  as  is  almost  universally 
done  nowadays  for  concrete  beams. 

Practical  formulas  for  unit  stresses  based  on  this  "  common 
theory"  cannot  be  deduced  for  this  case.  But  a  diagram 
can  be  constructed  by  means  of  which  unit  stresses  can  be 
easily  determined  for  any  section  of  a  given  chimney;  also  the 
amount  of  vertical  reinforcement  required  at  any  section  of 
a  given  concrete  shell  can  be  readily  determined  by  it.  Such 
a  diagram  will  now  be  described  by  example,  and  then  the 
analysis  on  which  its  construction  is  based  will  be  given.  At 
the  base  (Fig.  135)  there  are  given  values  of  "  eccentricity  " 


REINFORCED-CONCRETE    CHIMNEYS.         "  [Cn.  XI 


0.5 


Jfc 


lttm 


ffi 


45' 


15- 


10 


3Z 


1.0 


^alues  of  Eccentricity,  ^-j-[r 


/  / 


15 


y 


f^ 


X 


x 


2* 


0.5  1.0  1.5  2.0 

FIG.  135. — Wind  Stresses  in  Chimneys 


2.5 


§209.]  WIND    STRESSES.  409 

e/r,  and  at  the  left  side  values  of  m.  The  curves  relate  to  various 
percentages  of  steel,  from  0  to  4%,  and  will  be  called  percentage 
lines;  and  the  straight  lines  relate  to  various  ratios  of  fs/fc 
and  will  be  called  ratio  lines. 

Examples. — (1)  Ex.  (2),  Art.  208,  will  bemused  as  illustration.  From 
its  solution,  TF  =  275,000  Ibs.,  M  =  21,900,000  in-lbs., .  p  =  0.8%,  and 
4=2,636  in2  ;  hence  6  =  21,900,000/275,000  =  79.6  in.,  and  e/r  =  79.6/70 
=  1.14;  also  W/A  =  104  lbs/in2.  Now  entering  the  diagram  at  e/r  =  1.14, 
we  trace  vertically  upward  to  a  point  corresponding  to  an  0.8  percentrige- 
line,  and  then  horizontally  to  the  left  side,  taking  out  the  value  ra  =  3.65. 
We  also  note  that  the  turning  point  is  practically  at  the  20  ratio-line. 
Hence  fc  =  3.65X104  =  380  lbs/in2,  and  f8  =  20X380  =  7600  lbs/in2.  It 
may  be  noticed  that /r  is  not  the  unit  stress  at  the  remotest  fiber,  and 
hence  not  the  maximum  compressive  unit  stress  in  the  section.  The 
maximum  can  be  readily  computed  from 

/=/*+(/« +/./»)*/4r, 

in  which  t  denotes  thickness  of  the  concrete  shell.  (The  formula  may 
be  deduced  from  similar  triangles  in  the  lower  part  of  Fig.  136.)  Here 

/=380+  (380  +  500)  6/280=399  lbs/in2. 

(2)  How  much  vertical  reinforcement  is  needed  at  the  base  of  the 
chimney,  the  working  strengths  of  concrete  and  steel  being  limited 
to  500  and  15,000  lbs/in2  respectively? 

W  is  about  456,000  Ibs.,  the  wind  pressure  about  54,750  Ibs.,  and 
M  about  4,106,000  ft-lbs.;  hence  6  =  4,106,000/456,000  =  9  ft.  =  108  in., 
and  e/r  =  108/69  =  1.56.  The  section  area  A  is  3490  in2,  and  W/A 
=  130  lbs/in2;  hence  if  the  amount  of  steel  is  just  sufficient  to  make 
/c  =  500,  then  m  =  500/130  =  3.85.  Now  entering  the  diagram  at  e/r  = 
1.56  and  m  =  3.85,  we  trace  vertically  and  horizontally  from  these  places 
respectively,  and  note  the  intersection  at  about  p  =  1.9%  and /«//•  =  19. 
With  this  percentage  of  steel,  fs  =  19X500 =9500  lbs/in2.  (This  is  a 
low  working  stress ;  use  of  a  thicker  shell  will  make  higher  values 
possible  without  increase  of  amount  of  steel.  Several  trial  sections 
with  different  thicknesses  may  be  quickly  analyzed  by  means  of  the 
diagram,  and  an  economical  size  determined.) 

Analysis  for  the  Diagram.— The  ring  NPNQ  (Fig.  136) 
represents  a  section  of  a  chimney,  NN  the  neutral  axis,  and 


410 


REINFORCED-CONCRETE    CHIMNEYS. 


[On.  XI. 


NQN  the  compression  area,  the  wind  blowing  from  the  right. 
In  addition  to  the  foregoing  notation,  let 

Cc=  resultant  compresive  stress  in 

the  concrete; 
Cs  =  resultant  compressive  stress  in 

the  steel; 
T  =  resultant  tensile  stress  in  the 

steel  ; 

ac=arm  of  the  resultant  compres- 
sion Cc  +  Cs  with  respect  to 
the  center  0;  and 
a<=arm  of    the  resultant  tension 
T     with     respect    to    the 
FIG-  136'  center  0. 

The  resultant  normal  stress  on  the  section  equals  the  bending 
moment,  that  is, 

Ts  =  W      ......     (1) 


and 


(2) 


These  two  equations  constitute  the  basis  of  the  solution;  how- 
ever, they  must  be  modified  considerably,  and  this  will  be 
done  presently.  Let 

xi  =  distance  from  NN  to  the  centroid  of  the  arc  NPN; 
x2  =  distance  from  NN  to  the  centroid  of  the  arc  NQN', 
2/i  =  radius  of  gyration  with  respect  to  NN  of  arc  NPN] 
?/2  =  radius  of  gyration  with  respect  to  NN  of  arc  NQN',  and 

6  =  angle  NOP. 

Now  the  average  unit  compressive  stress  in  the  concrete  is 
/o£2/(l  +  cos  6)r,  that  in  the  compressive  steel  is  nfcX2/(l  +  cos  6)r, 
and  that  in  the  tensile  steel  is  nfcXi/(l  +  cos  6}r.  And  since 
the  area  of  the  section  of  the  compressive  concrete  is  practically 
—  d/x),  that  of  the  section  of  the  compressive  steel 
—  d/K),  and  that  of  the  tensile  steel  pAO/n,  it  follows  that 

0)r, 


§209.]  WIND    STRESSES.  411 

and  Ts = pA  (d/n)nfcXi/(l  +  cos  6)r. 

These  values  of  Cc,  C«,  and  Ts  substituted  in  eq.  (1)  give 

(1  —  6/x)(l  +  np)x2/r—  (0/Ti)np  x\/r—  (1  +  cos  0)W/Afc.     (3) 

From  any  source  of  information  on  the  centroid  of  a  circular 
arc,  it  can  be  shown  that 

/sin  6  \  /sin  6 

3i  =  r(— 5 cos  6}     and     x2  =  r{ 


\7T  — C7 


Imagining  these  values  substituted  in  eq.  (3),  it  will  be  seen 
that  the  left-hand  member  is  a  function  of  6,  n,  and  p  only. 
Denoting  this  function  by  Fi(6,  n,  p),  the  equation  can  be 
written  thus, 


c~Fi(0,n,p)  A' 
hence,  also, 


that  is,  m  depends  on  6,  n,  and  p  only. 

Referring  to  statement  in  the  preceding  article  about  the 
point  of  application  of  resultant  stress  and  to  Fig.  136,  it  will 
be  seen  that 

a*  =  2/i2Ai  +  r  cos  6. 

And  since  the  resultants  Cc  and  Cs  are  practically  equally 
distant  from  the  neutral  axis,  the  arms  of  Cc  +  C8  and  Cs  are 
practically  equal;  that  is,  approximately, 

cic=y22/X2  —  r  cos  6. 

If  now  these  values  of  at  and  ac  and  those  for  Cc+Cs  and  T9 
be  substituted  in  eq.  (2),  it  will  reduce  to 


(6) 


r    x2r 

W  e 


412 


REINFORCED-CONCRETE   CHIMNEYS. 


[Cn.  XI. 


It  can  be  shown  that  2/i2  =  r2[l  +  i  cos  20 -j  (sin  20) /0]  and 
2/22  =  r2[l  + J  Cos  20  +  f(sin  26) /(x— 6)].  Imagining  these  values 
of  t/i  and  3/2  substituted  in  eq.  (6),  it  will  be  seen  that  the 


1.5 

1.0 
0.5 

n 

7 

/ 

f 

Q 

u= 

15 

ind 

p=l 

/ 

/ 

ej 

xO 

// 

-1 

\ 

C/rf  J 

# 

[/ 

^ 

4> 

ffV 

^ 

£— 
1 

—  — 

,^-r 
-^—  • 

^*^^ 


..  —  - 

^^^- 

Values  of  9 
FIG.  137. 

left-hand  member  is  a  function  of  0,  n,  and  p  only.     Denoting 
this  function  by  F2(0,  n,  p),  the  equation  can  be  written, 

(l  +  cosd)(e/r)W/Afc=F2(6,n,p).      ...     (7) 
Division  of  eq.  (7)  by  eq.  (4)  gives 

e     F2(d,  n,  p) 

7=s:F1(etntpy 

that  is,  e/r  depends  on  0,  n,  arjd  p  only. 

Equations  (5)  and  (8)  are  the  desired  modifications  of 
eqs.  (1)  and  (2).  If  both  be  plotted  on  a  0  base  fora  given  set 
of  values  of  n  and  p  (n  =  15  and  p  =  0.01,  say),  a  pair  of  curves 
results  as  sketched  in  Fig.  137,  from  which  may  be  taken  the 
value  of  m  for  any  value  of  e/r.  Finally,  if  such  simultaneous 
values  of  m  and  e/r  (corresponding  to  one  and  the  same  value 
of  0)  be  taken  off  from  this  pair  of  curves  and  these  values 
be  plotted  on  an  e/r  base,  the  resulting  curve  is  the  1%  line 
of  the  diagram,  Fig.  135..  In  a  similar  manner  the  other 
percentage  lines  can  be  determined. 

The   ratio   lines   may   be   determined   as   follows:    From 

similar  triangles  in  the  lower  part  of  Fig.  136  it  is  plain  that 

fs/nfc=(l  —  cos  0)/(l  +  cos  0),    or 


§211.]  TEMPERATURE    STRESSES.  413 

From  this  equation  it  appears  that  for  /s//c  =  5,  say,  and 
n=15,  0  =  60°,  irrespective  of  the  value  of  p.  Now  the  values 
of  e/r  for  6  =  60°,  and  p  =  Q,  J,  1,  1J,  etc.,  may  be  read  off 
from  the  corresponding  pairs  of  curves  (like  Fig.  137),  and 
these  values  of  e/r  may  be  marked  off  on  the  corresponding 
percentage  curves  in  the  diagram;  the  points  so  marked  off 
fix  the  fa/fc  =  5  line.  In  a  similar  manner  the  other  ratio 
lines  can  be  determined. 

Since  kr  =  r—rcos6  (see  Fig.  136),  &  =  !(!  —  cos  6).  From 
this  formula  and  Fig.  137  the  value  of  k  may  be  obtained  for 
any  value  of  e/r  and  p=l%.  In  this  way  the  1%  curve  in 
Fig.  134  was  obtained;  and  in  a  similar  way  the  others. 

210.  Wind  Pressure. — Recent  experiments   made  on  the 
Eiffel  Tower  and  at  the  National  Physical  Laboratory  of  Eng- 
land show  that  the  pressure  per  square  foot  on  square  flat  sur- 
faces from  10  to  100  square  feet  in  extent  is  0.0032  times  the 
square  of  the  wind  velocity  in  miles  per  hour.     There  is  some 
evidence  that  the  pressure  on  a  cylindrical  surface  is  about  two- 
thirds  that  which  would  exist  on  an  axial  section  of  the  cylinder 
("projected  area").     On  the  basis  of  the  above,  20  pounds  per 
square  foot  of  projected  area  is  a  safe  value  for  chimneys.    The 
Prussian  Regulations  permit  use  of  17  pounds  per  square  foot; 
in  American  practice  considerably  higher  values  are  used. 

211.  Temperature   Stresses. — Fig.  138,   which   is    from  a 
photograph,  shows  plainly  some  large  vertical  and  horizontal 
cracks  in  the  outer  shell  of  a  chimney.     The  vertical  cracks 
are  doubtless  due  to  temperature  and  probably  the  horizontal 
ones  also.     For  since  the  inner  part  of  the  shell  is  hotter  than 
the  outer,  the  inner  tends  to  expand  more  circumferentially 
and  vertically  than  the  outer,  so  that  it  stretches  the  outer 
part  and  is  itself  compressed  vertically  and  circumferentially 
by  the  outer  part.     If  the  circumferential  or  vertical  tensions 
in  the  outer  part  are  excessive,  the  concrete    will  crack  on 
vertical  or  horizontal  planes  respectively. 

212.  Circumferential    Temperature    Stress.  —  The   following 
are    formulas    for  the  greatest   unit   compression  in  the  con- 


414 


REINFORCED-CONCRETE    CHIMNEYS. 


[Cn.  XI. 


crete   and    the   unit  tension   in  the  steel  at  any  place  in  a 
chimney  : 

c    and      s  = 


me  and  w«  being  certain  multipliers  which  depend  on    the 
percentage  of  hoop  reinforcement,  the  position  of  the  hoops 


FIG.  138. 

relative  to  outer  and  inner  surfaces  of  the  shell,  and  the  ratio 
of  outer  to  inner  radii.  For  meaning  of  the  other  symbols 
see  page  417.  Formulas  for  the  multipliers  are  complicated 
(page  419)  but  can  be  made  available  for  practical  use  by 
graphical  means.  They  have  been  plotted  in  a  diagram  (Fig. 


§212.] 


CIRCUMFERENTIAL   STRESSES 


415 


0.5 


0.4 


,0.3 


cO.2 


0.1 


\ 


\ 


\ 


\ 


For  the  solid  curves,      =   . 
u      u  dashed     «     ,  fc=0.4 


1.1 

1.2 
1.1 
1.2 

1.1. 

1.2 


Percentage  of  hoop  steel 
FIG.  139. 


416  REINFORCED-CONCRETE    CHIMNEYS..  [Cn.  XI 

139)  for  percentages  of  steel  from  0.2  to  2,  for  proportionate 
depths  of  steel  from  0.3  to  0.4,  and  for  ratios  of  outer  to  inner 
radii  from  1.1  to  1.2.  Inspection  of  the  diagram  shows: 

(1)  The  higher  the  percentage  of  steel,  the  lower  is  the 
unit  stress  in  the  steel  fs  and  the  higher  is  that  in  the  con- 
crete fc. 

(2)  Increasing  the  thickness  of  the  shell  within  practical 
limits  decreases  the  unit   stresses  in  concrete  and  steel  but 
not  materially,  less  than  10%. 

(3)  Moving  the  steel  inward  from  0.3t  to  0.4^  decreases 
both  unit  stresses,  from  10  to  20%  depending  upon  the  amount 
of  reinforcement. 

Example. — The  internal  diameter  of  a  chimney  shell  is  12  ft.,  the 
thickness  of  its  walls  is  6  in.,  the  hoops  are  J-in.  rounds  10  in.  apart, 
and  are  placed  so  that  the  center  of  the  steel  is  2  in.  from  the  outer 
surface  of  the  shell.  What  are  the  temperature  stresses  in  steel  and 
concrete  due  to  a  temperature  difference  of  200°  F.? 

The  percentage  of  steel  is  0.1964/ (6X10)  =0.0033  =  1.3%,  r2/r1  =  l.l, 
and  fc'  =  2/6  =  l/3.  For  fc'  =  0.3,  p  =  l%  and  r^r^l.l,  the  diagram 
gives  rac  =  0.18  and  ras  =  0.375;  for  /c'  =  l/3,  mc  =  0.19,  and  ras  =  0.40 
about.  Hence  if  K  =  0.000006,  Ec  =  2,000,000  and  Es  =  30,000, 000 

lbs/in2, 

fc  =  200  X  0.000006  X  2,000,000  X  0.19  =  456 

and  fa  =  200 X  0.000006  X  30,000,000  X  .40  =  14,400  lbs/in2. 

Analysis  for  the  Diagram. — In  Fig.  139,  MNPQ  represents 
a  portion  of  a  horizontal  section  of  a  chimney  and  0  the  center 
of  the  section.  The  "  neutral  line "  represents  the  neutral 
(cylindrical)  surface  which  is  not  stressed,  within  and  without 
which  the  concrete  is  under  compression  and  tension  respect- 
ively. The  distance  of  the  circumferential  steel  from  the 
outer  surface  of  the  shell  is  called  k't',  r  denotes  the  radius 
of  any  circumferential  "  fiber,"  C  the  total  compressive  stress 
on  the  vertical  section  MN  per  foot  of  height,  and  T  the 
total  tension  in  the  circumferential  steel  per  foot  of  height. 
Also  let 


$  212.]  CIRCUMFERENTIAL    STRESSES.  417 

t= thickness  of  concrete  shell  at  the  section  under  con- 
sideration ; 

7*1  =  inner  radius  of  shell; 

r2  =  outer  radius  of  shell; 

T= difference  of  temperatures  of  concrete  at  outer  and  inner 
faces; 

K = coefficient  of  expansion  of  concrete  and  steel; 

Ec  =  modulus  of  elasticity  of  concrete  in  compression; 

Es= modulus  for  steel; 

fc  =  temperature  unit  stress  in  concrete  (circumferential)  at 
inner  face;  and 

fs= temperature  unit  stress  in  circumferential  steel. 

In  this  analysis  the  tensile  value  of. the  concrete  is  neg- 
lected and  an  average  modulus  of  elasticity  for  concrete  in 
compression  is  assumed  for  all  unit  stresses,  just  as  in  com- 
putations on  the  strength  of  reinforced-concrete  beams.  The 
temperature  gradient  is  assumed  to  be  straight  and  the  coeffi- 
cients of  linear  expansion  for  concrete  and  steel  are  taken  as 
equal  and  constant  for  all  temperatures  involved  (see  Art.  28). 
Furthermore,  it  is  assumed  that  the  thickness  of  the  shell 
remains  unchanged  and  that  the  radii  of  all  circumferential 
fibers  are  increased  equally.  (Strictly  this  is  not  true  for 
there  are  radial  expansions  and  contractions  accompanying 
the  circumferential  stresses,  and  then  there  is  radial  shorten- 
ing accompanying  the  radial  compressive  stress;  but  these 
are  small  and  their  observance  is  out  of  place  in  this  analysis 
involving  as  it  does,  approximation  of  a  larger  order.  On 
account  of  the  unequal  vertical  expansions  there  will  be  cir- 
cumferential expansion  at  the  top,  the  chimney  "  belling " 
out  there,  and  some  contraction  lower  down.  In  the  follow- 
ing analysis,  these  are  neglected;  the  resulting  error  is  prob- 
ably small  except  for  stresses  near  the  top.  The  assumption 
that  the  modulus  of  elasticity  is  constant  for  the  range  in 
temperature  may  be  quite  erroneous,  and  if  so,  the  analysis 
following  is  in  error;  if  the  modulus  is  lower  for  the  higher 


418  REINFORCED-CONCRETE    CHIMNEYS.  [CH.  XI. 

temperatures  then  the  actual    unit    stresses    are  lower  than 
those  here  found.) 

If  Ar  denotes  the  radial  increase  mentioned,  then  the  actual 
elongation  of  each  circumference  is  2xJr,  and  the  actual  unit 
elongation  is  Ar/r.  The  temperature  at  the  distance  r  from 
the  center  (reckoned  from  the  temperature  at  the  outer  surface 
as  zero)  is  r(r2  —  r)/£,  and  hence  the  free  elongation  of  the 
circle  of  radius  r  would  be  2xrKT(r2  —  r)/t  and  the  free  unit 
elongation  Kr(r2  —  r)/t.  The  difference  between  the  free  and 
the  actual  unit  elongations  is  the  prevented  unit  elongation, 
and  hence  the  corresponding  preventing  unit  stress  in  the 
concrete  is 


;    .....    (1) 
for  the  steel,  r  becomes  equal  to  r2  —  k't  and 


At  the  neutral  surface  /=0,  and  r  =  ri  +  kt;    hence  on  substi- 
tuting these  in  eq.  (1),  it  will  be  found  that 


(3) 


A  value  of  the  compression  at  the  inner  face  may  be  obtained 
from  (1)  by  substituting  for  r  its  value  there  and  for  Ar  its 
value  from  eq.  (3)  ;  the  expression  will  reduce  to 

.     (1)' 


A  new  value  of  fs  can  be  obtained  from  (2)  by  substituting 
for  Ar  its  value  from  eq.  (3)  ;  it  will  reduce  to 


These  expressions  for  /«.  and  /«  contain  only  quantities  ordi- 
narily known  and  k;  it  remains  to  determine  k.  This  is  done 
by  means  of  the  condition  that  C  and  T  are  equal.  Now, 


§  213.]  VERTICAL   TEMPERATURE   STRESS.  419 

r+kt 
fl2dr    and    T=fspl2t;    and  if  in  these,  the  values 

of  /  and  fs  from  eqs.  (1)  and  (2)  be  inserted,  and  the  resulting 
expressions  for  0  and  T  be  equated,  and  then  if  the  operations 
indicated  in  the  equation  be  performed,  the  following  may 
be  arrived  at: 

Jr  =  Krt(k  -  \W>  +  npkf)  (r2/t  -  V)  +  [np  +  (r2/t  -  A/)log(l  +  Vt/n)} 

The  logarithm  is  Naperian,  or  natural.  Equating  this  value 
of  Jr  to  that  given  by  eq.  (3)  and  simplifying,  one  may  arrive  at 


Now  tliis  equation  determines  k  for  given  values  of  k',  n,  and 
r2/ri,  and  yet  it  cannot  be  solved  for  k  on  account  of  the 
logarithmic  term.  However,  values  of  p  can  be  determined 
for  given  values  of  k,  k',  n,  and  7*2/7*1,  and  thus  a  diagram 
can  be  made  like  the  group  of  k  curves  (Fig.  139)  but  with 
much  larger  vertical  scale,  from  which  k  may  be  taken  off 
for  any  given  case  (p,  k',  n,  r2/ri).  Then  this  value  may  be 
used  in  eqs.  (1)'  and  (2)'  to  determine  the  bracketed  coeffi- 
cients, that  is,  me  and  ms. 

213.  Vertical  Temperature  Stress.  —  A  satisfactory  analysis 
is  not  available;  the  following  approximation  indicates  the 
"  order  of  magnitude  "  of  these  stresses.  Horizontal  sec- 
tions through  the  unheated  chimney  are  still  horizontal  after 
heating  just  as  plane  sections  of  a  beam  remain  plane  during 
bending.  Hence  the  inner  part  of  the  chimney  shell  when 
hot  is  compressed  while  the  outer  part  is  stretched,  and  some- 
where between  there  will  be  a  neutral  surface  whose  distance 
from  the  inner  surface  is  here  called  kt  (see  Fig.  140).  If 
the  tensile  strength  of  the  concrete  is  neglected,  then  the 
entire  vertical  tension  must  be  ascribed  to  the  steel,  and  the 
neutral  surface  located  between  the  steel  and  inner  surface 
as  shown. 


420 


REINFORCED-CONCRETE    CHIMNEYS. 


[Cn.  XI. 


The  temperature  difference  between  the  inner  surface  and 
the  neutral  surface  is  rk,  and  the  temperature  difference 
between  the  neutral  surface  and  the  steel  is  r(J  —  &);  hence 
the  unit  stress  at  the  inner  surface  fc,  and  the  unit  stress  in 
the  steel/*  are  given  by 

fc=krKEc,     .    .    .    .    . '-'.••  .'-.     (1) 
and  fs-a-VrKE*.      .    .    v.    .    ,    (2) 

To  determine  k  equate  the  total  compression  and  the  total 
tension  per  unit  of  circumference;  thus  p  denoting  the  (ver- 
tical) "  steel  ratio/ '  or  total  vertical  steel  area  divided  by 


FIG.  140. 


total    area    of    cross-section    of    the    shell,  then    ^ 
(\-k)i:KEspt,    which    simplified    and    solved    for    k    gives 


From  this  equation  the  following  table  was  computed,  n  taken 
as  15: 


p= 

0.05 

0.10 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

k  = 

0.21 

0.26 

0.30 

0.32 

0.34 

0.36 

0.37 

0.38 

For  2%  of  steel,  r=200°,  and  K= 0.000006,  formulas  (1) 
and  (2)  give  fc= 770  and/,  =  6500  lbs/in2. 


$  215.]  BASES.  421 

2 14.  Chimney  Temperatures. — Sanford,  in  the  report  alluded 
to  in  Art.  206,  states  that  "the  temperature  in  an  ordinary  chimney 
seldom  exceeds  700°  F.  at  the  base  and  400  to  500  is  more  usual." 

Whatever  the  difference  between  the  temperatures  of  the 
chimney  gas  and  the  outer  air,  the  difference  r  between  the 
temperatures  of  the  concrete  at  the  inner  and  outer  face  of 
the  chimney  shell  (on  which  the  temperature  stresses  depend) 
is  less,  for  it  is  known  that  at  the  surfaces  of  a  heat  barrier 
there  is  a  drop  in  temperature  in  the  direction  of  the  heat 
flow-  And  at  the  outer  surface  of  a  chimney  the  drop  is  con- 
siderable, as  is  known  to  any  one  who  has  placed  his  hand 
upon  the  surface  on  a  cold  day;  to  him  it  felt  warm  while 
the  air  temperature  may  have  been  zero  or  less.  Lange*  has 
computed  the  temperature  drops  at  the  surfaces  of  some  brick 
chimneys  (various  diameters,  thickness  of  walls,  gas,  and  air 
temperatures)  and  while  the  computations  seem  to  be  based 
on  uncertain  values  of  thermal  conductivity,  emissivity,  and 
absorption  of  the  chimney  material,  still  they  are  doubtless 
reliable  enough  to  indicate  that  the  temperature  difference 
for  a  concrete  shell  may  be  as  little  as  50%  of  the  temperature 
difference  for  the  chimney  gases  and  air. 

215.  Bases. — In  the  two  succeding  articles  there  are  ex- 
plained   methods    for    computing    the    maximum    pressures 
between  the  base  of  a  given  chimney  and  its  earth  or  stone 
foundation,  and  the  strength  of  the  base  to  withstand  those 
pressures  and  the  forces  on  its  top.     The  notation  is  as  follows 
(see  also  Figs.  141  and  142) : 

W= total  weight  of  chimney  and  earth  filling  over  the  base; 
M=wind  moment  at  the  bottom  of  the  base; 
A — area  of  the  bottom  of  the  base ; 
p2  =  maximum  unit  pressure  on  bottom; 
pi  =  minimum  unit  pressure  on  bottom; 
r  =  kern  radius  of  bottom  in  direction  of  wind; 
e  =  eccentricity  at  bottom  of  resultant  of  the  wind  pressure 
and  W,  e  =  M/W. 

*  Der  Schornsteinbau. 


422 


REINFORCED-CONCRETE  CHIMNEYS. 


[Cn.  XI. 


216.  Earth  Pressures. — It  is  assumed  that  the  pressure  is  a 
uniform  y  varying  one.  If  the  resultant  of  the  wind  pressure 
on  the  chimney,  the  weight  of  the  chimney  and  earth  filling 
over  the  base  cuts  the  kern  of  the  bottom  of  the  base,  then 
there  will  be  no  tendency  for  the  windward  toe  to  lift,  or,  in 
other  words,  there  will  be  no  tendency  to  tension  between 
the  earth  or  rock  foundations  and  the  windward  toe.  Fig.  141 
shows  the  kerns  for  square,  octagonal,  and  circular  bottoms. 
It  is  good  practice  to  make  the  bottom,  in  a  given  case,  large 
enough  so  that  its  kern  will  intercept  the  line  of  action  of 


FIG.  141. 

the  resultant  R;  however,  this  is  not  an  absolutely  essential 
requisite  for  stability  and  safety. 

//  the  resultant  R  cuts  the  kern  then  the  computation  for  the 
greatest  and  least  unit  pressure  is  comparatively  simple.  These 
are  the  formulas : 

W    M  W    M 

Pi=-r  +  -r-    and     7)2  =  -r—  T— .        .     .     .     (1) 
A     Ar  A     Ar 

Evidently  the  greater  pressure  pi  is  a  maximum  when  the 
kern  radius  is  minimum;  hence  pi  is  maximum  when  the 
wind  pressure  is  parallel  to  the  longest  diameter  of  the  base. 

//  the  resultant  R  does  not  cut  the  kern  then  there  is  a 
neutral  axis  as  it  were,  that  is,  only  a  part  of  the  bottom  is 
under  pressure.  This  neutral  axis  is  perpendicular  to  the 
direction  of  the  wind  pressure.  Three  cases  are  noted : 

(a)  Square  Bases. — If  the  direction  of  the  wind  is  parallel 


§  216.]  "*        EARTH    PRESSURES.  423 

to  a  side  of  the  square/  then  the  distance  x  of  the  neutral 
axis  from  the  windward  of  the  square  is 


and  the  greatest  unit  pressure  is 

4          W 

Pl  =  3(l-2e/b)  W' 


(2)1 


These  formulas  follow  from  two  equations  obtained  by  equat-  j 
ing  the  total  pressure  on  the  bottom  to  W,  and  the  moment  | 
of  that  pressure  about  the  diameter  parallel  to  the  neutral 
axis  to  M=We. 

If  the  direction  of  the  wind  is  parallel  to  a  long  diameter 
of  the  square,  then  the  position  of  the  neutral  axis  (see  Fig. 
142)  is  given  by 


and  the  value  of  the  greatest  unit  pressure  by 

2-k      W 


In  Fig.  142  there  are  two  curves  marked  "  square;  "  one 
gives  values  of  k  and  the  other  values  of  m  for  use  in  pi  =mW/b2. 
Eqs.  (3)  and  (4)  were  deduced  from  the  same  principles  em- 
ployed to  deduce  (1)  and  (2). 

(6)  Octagonal  Bases.  —  Exact  formulas  are  not  practical. 
Since  an  octagon  does  not  differ  much  from  a  co-centric  circle 
whose  diameter  equals  the  mean  of  the  greatest  and  least 
diameters  of  the  octagon,  it  must  be  that  the  neutral  axis 
and  the  greatest  unit  pressure  for  an  octagonal  bottom  do 
not  differ  materially  from  those  for  such 

(c)  Circular  Bases.  —  In  Fig.  142  there  are  two  curves  marked 
"  circle;  "  one  of  these  gives  values  of  k  and  the  other  values 


424 


REINFORCED-CONCRETE   CHIMNEYS. 


[Cn.  XI. 


of  m  for  use  in  pi  =  mW/A,  A  denoting  area  of  the  circle  and 
pi  the  greatest  unit  pressure  on  it. 


4.0 


2.0 

0.8 


0.0 


0.4 


0.2 


0.1 


0.2  0.3  0.4  0.5 

Values  of  eccentricity  %• 

FIG.  142. 


217.  Design  of  Bases. — Having  determined  the  diamete 
of  the  base  from  a  consideration  of  the  earth  pressures  as 
explained  in  the  preceding  article,  it  rema'ns  to  determine 
thicknesses  of  the  bases  and  the  reinforcement.  Only  rough 
approximate  methods  are  available.  A  chimney  base  is  essen- 
tially a  column  footing  for  methods  of  the  design  of  which 
see  Art.  169.  But  while  column  footings  are  regarded  as 
always  subjected  to  a  uniform  earth  pressure,  a  chimney  base 
should  not  be  so  regarded;  and  the  outstanding  cantilever 
part  of  a  base  should  be  figured  for  the  earth  pressure,  dis- 
tribution obtaining  with  the  maximum  wind  pressure  as 
explained  in  the  preceding  article.  Also  while  the  column 


§217.]  DESIGN    OF   BASES.  425 

is  solid,  the  chimney  is  not;  and  in  the  latter  case  tensile 
stresses  may  arise  at  the  top  side  of  the  base  at  its  center. 
Such  tension  will  probably  obtain  when  the  inner  diameter 
of  the  chimney  is  greater  than  twice  the  length  of  the  out- 
standing cantilever  and  no  wind  blowing. 


INDEX. 


Adhesion,  33. 

of  concrete  and  reinforcing  bars,  33 

tests  of,  34 

working  values  of,  216 
Analysis  of  arches,  335-352 
Anchored  bars,  217 
Arches, 

advantages  of  reinforced,  6,  333 

analysis  of,  335 

deflection  of,  344 

examples  of,  363 

methods  of  reinforcing,  334 

stresses  in,  336,  351 

temperature  stresses  in,  342 

unsymmetrical,  344 

Bars, 

forms  of,  30 
spacing  of,  226 

Beams, 
advantages  of  reinforced  concrete 

for,  5 

applicability  of  theory  of,  155,  211 
compression  reinforcement  in,  173 
continuous,  301,  319 
deflection  of,  116,  176 
depth  of,  239 
diagonal  tension  failures  of,  155, 

163 

diagrams  for  simple,  275 
diagrams  for  T-beams,  90,  280 
double  reinforced,  92,  174 
double  reinforcement  of,  173 
economical  proportions  of,  227 
flexure  and  direct  stress  in,  98,  266 
formulas  for  (see  Formulas) 
methods  of  failure  of,  132 
shear  failures  of,  155,  163 
shear  reinforcement  for,  157 
shearing  stresses  in,  108,  157,  269 
T-beams  (see  T-beams  ) 
tension  failures  of,  141 
tests  of,  142,  163 
variation  of  stress  in,  47 
web  reinforcement  in,  157,  219 
working  stresses  in,  212 


Bins,  396 
Bond,  33 

mechanical,  35 

tests  of,  34 
Bond  stress,  113,  269 

working  values  of,  216 
Bridges  (see  Arches  and  Girders) 
Broken  stone,  general  requirements,  9 
Building  construction,  301 

advantages  of  reinforced  concrete 


Cement,  general  requirements,  9 
Chimneys,  advantages  of  reinforced 
concrete  for,  7 

bases  of,  421 

linings  of,  399 

temperature  stresses  in,  413 

temperatures  of,  421 

wind  pressure  on,  413 

wind  stresses  in,  400 
Cinder  concrete,  properties  of,  29 
Coefficient  of  expansion  of 

concrete,  26 

steel,  33 
Columns, 

advantages  of  reinforced  concrete 
for,  5 

details  of,  250,  324 

diagram  for,  289 

eccentric  loads  on,  323 

economy  of  reinforced  concrete  for, 
252 

examples  of  design  of,  324 

formulas  for,  128,  131 

hooped,  131,  193 

notation  for,  127,  270 

reinforcement  of,  127 

tests  of  plain  concrete,  184 

tests  of  reinforced,  187,  193 

working  stresses  for,  243 
Concrete, 

cinder,  29 

coefficient  of  expansion  of,  26 

consistency  of,  10,  13 

crushing  strength  of,  11 

427 


428 


INDEX. 


Concrete, 

in  beams,  153 

elastic  limit  of,  25 

elastic  properties  of,  19 

elongation  of,  37 

expansion  of,  26,  27 

fatigue  tests,  206 

general  requirements,  8 

modulus  of  elasticity  of,  20,  40 

proportions  of  ingredients  of,  10 

temperature  stresses  in,  44 

tensile  strength  of,  15 

transverse  strength  of,  16 

shearing  strength  of,  17 

shrinkage  of,  27 

shrinkage  stresses  in,  44,  256 

stress-strain  curves  of,  26 

variation  in,  9 

weight  of,  29 

working  stresses  for,  212 
Conduits,  395 

advantages  of  reinforced  concrete 

for,  7 

Continuous  beams,  301,  319 
Culverts, 

advantages  of  reinforced  conrcete 
for,  6 

examples  of,  390 

reinforcement  for,  390 

stresses  in,  381 


Dams,  383 

advantages  of  reinforced  concrete 

for,  6 

Deflection  of  beams,  116,  176 
Diagrams  for, 

circular  slabs,  290,  291 

columns,  289 

double  reinforcement,  286 

flexure  and  direct  stress,  287,  288 

simple  beams,  275 

T-beams,  90,  280 
Double  reinforcement,  diagram  for, 

286 
Double  reinforcement  of 

beams,  92,  173 

T-beams,  92 

Eccentric  column  loads,  323 

Factor  of  safety,  208 

Fatigue  tests,  206 

Fireproofing,  reinforced  concrete  for, 

255 

Flexure  and  direct  stress,  98,  266 
diagram  for,  287,  288 


Floors,  301 

design  of,  312 

examples  of,  324 
Floor-beams, 

arrangement  of,  316 

continuous,  301 

loads  on,  317 
Floor-slabs, 

continuity  of,  304 

design  of,  312 

reinforced  in  two  directions,  308 

shrinkage  reinforcement  for,  312 

tables  for,  295 
Footings,  330 
Formulas  for 

beams,  compared,  81 

beams,  double-reinforced,  92,  264 

beams,  Talbot's,  73 

beams,  ultimate  loads,  66,  69,  241 

beams,  variety  of,  56 

beams,  working  loads,  59,  69,  211. 
239 

columns,  128,  131 

T-beams,  82,  263 
Frictional  resistance  of  bars,  35 

Girders  (see  Beams) 
Girder  bridges,  385 
Gravel,  general  requirements,  9 

Historical  sketch,  1 
Hooks,  efficiency  of,  40 
Live  load,  effect  of,  209 

Mechanical  bond,  39 
Melan  system,  1 
Modulus  of  elasticity  of 

concrete,  20,  40 

steel,  32 

Monier  system,  1 
Mushroom  system,  329 

Neutral  axis,  146 
Notation  for 

beams,  59,  239 

beams,  double-reinforced,  92,  264 

columns,  128,  270 

flexure  and  direct  stress,  99,  267 

T-beams,  82,  263 

Piles,  use  of  reinforced  concrete  for,  7 
Pipes,  387,  390 
Plates,  circular, 

diagrams  for,  290,  291 

stresses  in,  272 


INDEX. 


429 


Railroad  ties,  use  of  reinforced  con- 
crete for,  7 
Ransome  system,  2 
Reinforced  concrete, 

advantages  of,  4 

durability  of,  253 

fire  resisting  qualities  of,  255 

history  of,  1 

repeated  load  tests  of,  206 

shrinkage  stresses  in,  40 

temperature  stresses  in,  44 

use  of,  4 

working  stresses  for,  208,  273 
Reservoirs,  396 

advantages  of  reinforced  concrete 

for,  6 
Retaining  walls, 

advantages  of  reinforced  concrete 
for,  6,  370 

design  of,  376 

examples  of,  380 

fluid  pressure  on,  371 

proportions  of,  375 

stability  of,  370,  373 

supported  at  top,  382 
Rods, 

spacing  of,  226 

tables  for,  292,  293 

Sand, 

effect  of  size  of  grains,  14 

general  requirements,  9 
Shear  failure  of  beams,  155,  163 
Shear  reinforcement  of  beams,  157 
Shearing  strength  of 

concrete,  17 

T-beams,  168 
Shearing  stress, 

effect  of,  on  tensile  stress,  110 

in  beams,  108,  157,  269 

working  values  of,  219 
Shrinkage  of  concrete,  27 
Shrinkage  stresses,  44,  256,  312 
Spacing  of  rods  or  bars,  226 
Steel, 

coefficient  of  expansion  of,  33 

corrosion  of,  in  concrete,  253 


Steel, 

elongation  of,  32 

general  requirements,  30 

modulus  of  elasticity  of,  32 

quality  of,  32,  215 

tensile  strength  of,  32 

working  stresses  for,  212 
Stirrups,  160 
Stress-strain  curves,  26 

Tables  for 

floor-slabs,  295 

rods,  292,  293 

Tanks,  advantages  of  reinforced  con- 
crete for,  7 
T-beams, 

deflection  of,  122,  176 

diagrams  for,  90,  280 

double  reinforced,  92 

economical  proportions  for,  237 

formulas  for,  83,  263 

notation  for,  83,  263 

shearing  strength  of,  168 

strength  of,  167 

tests  of,  167 
Temperature  stresses,  44,  256 

in  arches,  342 
Tests  of 

adhesion,  34 

baems,  142,  163,  174 

columns,  184,  187,  193 

T-beams,  167,  176 
Trestles, 

analysis  of  stresses  in,  386 

examples  of,  390 

Unit  frames,  327 

Walls,  332 

Web  reinforcement,  157,  219 
Working  stresses,  208,  273 
Working  stresses  for, 

beams,  212 

bond,  216 

columns,  243,  289 

concrete,  212 

steel,  212 


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